17254093 Quantitative

From the desk of SOHAIL MERCHANT__________________________________________________________________________________________ - 1 Quantitative Aptitude INDEX ARITHMETIC 1. Real Number System --------------------------------------- 03-12 2. *Ratio & Proportion --------------------------------------- 13-19 3. **Percentages --------------------------------------- 20-24 4. *Averages & Mixtures --------------------------------------- 25-33 5. **Profit & Loss --------------------------------------- 34-38 6. **Simple & Compound Interests --------------------------------------- 39-43 7. *Time & Work --------------------------------------- 44-49 8. *Time & Distance --------------------------------------- 50-56 9. *Age Problems --------------------------------------- 57-61 GEOMETRY 10. Plane Geometry --------------------------------------- 62-80 11. **Mensuration --------------------------------------- 81-91 COUNTING METHODS & PROBABILITY 12. Set Thoery --------------------------------------- 91-97 13. *Permutations & Combinations --------------------------------------- 98-103 14. *Probability --------------------------------------- 104-110 15. Progressions --------------------------------------- 111-117 16. Statistics --------------------------------------- 118-124 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 2 ALGEBRA 17. Progressions -------------------------- 126-132 18. Matrices -------------------------- 133-142 19. Statements -------------------------- 143-148 20. Sets -------------------------- 149-154 21. Real Numbers, Rational Numbers & Law of Indices -------------------------- 155-160 22. Surds -------------------------- 161-168 23. Linear Equations, Inequations & Modulus -------------------------- 169-174 24. Polynomials, Remainder & Square Roots -------------------------- 175-177 25. Quadratic Equations & Expressions -------------------------- 178-180 26. Relations & Functions -------------------------- 181-183 27. Derivatives & Limits -------------------------- 184-187 28. Logarithms -------------------------- 188-190 29. Binomial Theorem -------------------------- 191-193 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 3 NUMBER SYSTEMS In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. 0 is called insignificant digit. A group of figures, denoting a number is called a numeral. For a given numeral, we start from extreme right as Unit‘s place, Ten‘s place, Hundred‘s place and so on. Illustration 1 We represent the number 309872546 as shown below: T e n C r o r e 1 0 8 C r o r e s 1 0 7 T e n L a c s ( m i l l i o n ) 1 0 6 L a c s 1 0 5 T e n T h o u s a n d 1 0 4 T h o u s a n d 1 0 3 H u n d r e d 1 0 2 T e n ‘ s 1 0 1 U n i t s 1 0 0 3 0 9 8 7 2 5 4 6 We read it as ―Thirty crores, ninety- eight lacs, seventy-two thousands five hundred and forty-six.‖ In this numeral: The place value of 6 is 6 ×1 = 6 The place value of 4 is 4 ×10 = 40 The place value of 5 is 5 ×100 = 500 The place value of 2 is2 ×1000 = 2000 and so on. The face value of a digit in a numbers is the value itself wherever it may be. Thus, the face value of 7 in the above numeral is 7. The face value of 6 in the above numeral is 6 and in the above numeral is 6 and so on. NUMBER SYSTEM Natural numbers Counting numbers 1, 2, 3, 4, 5,... are know as natural numbers. The set of all natural numbers, can be represented by N= {1, 2, 3, 4, 5,….} Whole numbers If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5, … are called whole numbers. The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5…} Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number. INTEGERS All counting numbers and their negatives including zero are know as integers. The set of integers can be represented by Z or I = {…-4, -3, -2, -1, 0, 1, 2, 3, 4, …} Positive Integers The set I + ={1, 2, 3, 4,…} is the set of all positive integers. Clearly, positive integers and natural numbers are synonyms. Negative Integers The set I - = {-1, -2, -3…} is the set of all negative integers. 0 is neither positive nor negative. Non-negative Integers The set {0, 1, 2, 3,…} is the set all non-negative integers. Rational Numbers The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers, e.g. 4/7, 3/2, - 5/8, 0/1, -2/3, etc. The set of all rational numbers is denoted by Q. i.e. Q ={x:x =p/q; p,q belong to I, q≠0}. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 4 Since every natural number ‗a‘ can be written as a/1, every natural number is a rational number. Since 0 can be written as 0/1 and every non-zero integer ‗a‘ can be written as a/1, every integer is a rational number. Every rational number has a peculiar characteristic that when expressed in decimal form is expressible rather in terminating decimals or in non-terminating repeating decimals. For example, 1/5 =0.2, 1/3 = 0.333…22/7 = 3.1428704287, 8/44 = 0.181818…., etc. The recurring decimals have been given a short notation as 0.333…. = 0.3 4.1555… = 4.05 0.323232…= 0.32. Irrational Numbers Those numbers which when expressed in decimal from are neither terminating nor repeating decimals are known as irrational number, e.g. √2, √3, √5, π, etc. Note that the exact value of t is not 22/7. 22/7 is rational while π irrational number. 22/7 is approximate value of π. Similarly, 3.14 is not an exact value of it. Real Numbers The rational and irrational numbers combined together are called real numbers, e.g.13/21, 2/5, -3/7, \3, 4 + \2, etc. are real numbers. The set of all real numbers is denote by R. Note that the sum, difference or product of a rational and irrational number is irrational, e.g. 3+ √2, 4-√3, 2/3-√5, 4√3, -7√5 are all irrational. Even Numbers All those numbers which are exactly divisible by 2 are called even numbers, e.g.2, 6, 8, 10, etc., are even numbers. Odd Numbers All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1, 3, 5, 7 etc., are odd numbers. Prime Numbers A natural number other than 1 is a prime number if it is divisible by 1 and itself only. For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers. Composite Numbers Natural numbers greater than 1which are not prime, are known as composite numbers. For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers. Note: 1. The number 1 is neither a prime number nor composite number. 2. 2 is the only even number which is prime 3. Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100. 4. two numbers which have only 1 as the common factor are called co-primes or relatively prime to each other, e.g. 3 and 5 are co-primes. Note that the numbers which are relatively prime need not necessarily be prime numbers, e.g. 16 and 17 are relatively prime although 16 is not a prime number. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 5 ADDITION AND SUBTRACTION (SHORT-CUT METHODS) The method is best illustrated with the help of following example: Illustration 2 54321 – (9876+8976+7689) = ? Step 1 Add 1 st column: 54321 9876 8967 7689 27789 6+7+9 = 22 To obtain 1 at unit‘s place add 9 to make 31. In the answer, write 9 at unit‘s place and carry over 3. Step 2 Add 2 nd column: 3+7+6+8=24 To obtain 2 at tens place add 8 to make 32. In the answer, write 8 at ten‘s place and carry over 3. Step 3 Add 3 rd column: 3 + 8 + 9 + 6 = 26 To obtain 3 at hundred‘s place, add 7 to make 33. In the answer, write 7 at hundred‘s place and carry over 3. Step 4 Add 4 th column: 3 + 9 + 8 + 7 = 27 To obtain 4 at thousand‘s place add 7 to make 34. In the answer, write 7 at thousand‘s place and over 3. Step 5 5 th column: To obtain 5 at ten-thousand‘s place add 2 to it to make 5. In the answer, write 2 at the ten-thousand‘s place. 54321 – (9876 + 8967 + 7689) = 27789. Common Factor A common factor of two or more numbers is a number which divides each of them exactly. For example, 4 is a common factor of 8 and 12. Highest common factor Highest common factor of two or more numbers is the greatest number that divides each one of them exactly. For example, 6 is the highest common factor of 12, 18 and 24. Highest Common Factor is also called Greatest Common Divisor or Greatest Common Measure. Symbolically, these can be written as H.C.F. or G.C.D. or G.C.M., respectively. Methods of Finding H.C.F. I. Method of Prime Factors Step 1 Express each one of the given numbers as the product of prime factors. [A number is said to be a prime number if it is exactly divisible by 1 and itself but not by any other number, e.g. 2, 3, 5, 7, etc. are prime numbers] Step 2 Choose Common Factors. Step 3 Find the product of lowest powers of the common factors. This is the required H.C.F. of given numbers. Illustration 1 Find the H.C.F. of 70 and 90. Solution 70 = 2 × 5 × 7 90 = 2 × 5 × 9 Common factors are 2 and 5. H.C.F. = 2 × 5 = 10. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 6 Illustration 2 Find the H.C.F. of 3332, 3724 and 4508 Solution 3332 = 2 × 2 × 7 × 7 × 17 3724 = 2 × 2 × 7 × 7 × 19 4508 = 2 × 2 × 7 × 7 × 23 H.C.F. = 2 × 2 × 7 × 7 = 196. Illustration 3 Find the H.C.F. of 360 and 132. Solution 360 = 2 3 × 3 2 × 5 132 = 2 2 × 3 1 × 11 H.C.F. = 2 2 × 3 1 × = 12. Illustration 4 If x = 2 3 × 3 5 × 5 9 and y = 2 5 × 3 7 × 5 11 , find H.C.F. of x and y. Solution The factors common to both x and y are 2 3 , 3 5 and 5 9 . H.C.F. = 2 3 × 3 5 × 5 9 . II. Method of Division A. For two numbers: Step 1 Greater number is divided by the smaller one. Step 2 Divisor of (1) is divided by its remainder. Step 3 Divisor of (2) is divided by its remainder. This is continued until no remainder is left. H.C.F. is the divisor of last step. Illustration 5 Find the H.C.F. of 3556 and 3444. 3444 )3556 (1 3444 112 ) 3444 ( 30 3360 84 ) 112 ( 1 84 28 ) 84 ( 3 84 × B. For more than two numbers: Step 1 Any two numbers are chosen and their H.C.F. is obtained. Step 2 H.C.F. of H.C.F. (of(1)) and any other number is obtained. Step 3 H.C.F. of H.C.F. (of (2)) and any other number (not chosen earlier) is obtained. This process is continued until all numbers have been chosen. H.C.F. of last step is the required H.C.F. Illustration 6 Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm. Solution Required length = (H.C.F. of 700, 385, 1295) cm = 35 cm. Common Multiple A common multiple of two or more numbers is a number which is exactly divisible by each one of them. For Example, 32 is a common multiple of 8 and 16. 8 × 4 = 32 16 × 2 = 32. Least Common Multiple The least common multiple of two or more given numbers is the least or lowest number which is exactly divisible by each of them. For example, consider the two numbers 12 and 18. Multiples of 12 are 12, 24, 36, 48, 72, … From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 7 Multiple of 18 are 18, 36, 54, 72, … Common multiples are 36, 72, … Least common multiple, i.e. L.C.M. of 12 and 18 is 36. Methods of Finding L.C.M. A. Method of Prime Factors Step 1 Resolve each given number into prime factors. Step 2 Take out all factors with highest powers that occur in given numbers. Step 3 Find the product of these factors. This product will be the L.C.M. Illustration 7 Find the L.C.M. of 32, 48, 60 and 320. Solution 32 = 25 × 1 48 = 24 × 3 60 = 22 × 3 × 5 320 = 26 × 6 L.C.M. = 26 × 3 × 5 = 960. B. Method of Division Step 1 The given numbers are written in a line separated by common. Step 2 Divide by any one of the prime numbers 2, 3, 5, 7, 11, … which will divide at least any two of the given nu8mbers exactly. The quotients and the undivided numbers are written in a line below the first. Step 3 Step 2 is repeated until a line of numbers (prime to each other) appears. 1 Find the product of all divisors and numbers in the last line which is the required L.C.M. Illustration 8 Find the L.C.M. of 12, 15, 20 and 54. Solution 2 12, 15, 20, 54 2 6, 15, 10, 27 3 3, 15, 5, 27 5 1, 5, 5, 9 1, 1, 1, 9 L.C.M. = 2 × 2 × 3 × 5 × 1 × 1 × 1 × 9 = 540. Note: Before finding the L.C.M. or H.C.F., we must ensure that all quantities are expressed in the same unit. Some Useful Short-Cut Methods 1. H.C.F. and L.C.M. of Decimals Step 1 Make the same number of decimal places in all the given numbers by suffixing zero(s) if necessary. Step 2 Find the H.C.F./L.C.M. of these numbers without decimal. Step 3 Put the decimal point (in the H.C.F./L.C.M. of step 2) leaving as many digits on its right as there are in each of the numbers. 2. L.C.M. and H.C.F. of Fractions L.C.M = L.C.M. of the numbers in numerators H.C.F. of the numbers in denominators H.C.F. = H.C.F. of the numbers in numerators L.C.M. of the numbers in denominators 3. Product of two numbers = L.C.M. of the numbers × H.C.F. of the numbers From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 8 4. To find the greatest number that will exactly divide x, y and z. Required number = H.C.F. of x, y and z. 5. To find the greatest number that will divide x, y and z leaving remainders a, b and c, respectively. Required number = H.C.F. of (x – a), (y – b) and (z – c). 6. To find the least number which is exactly divisible by x, y and z. Required number = L.C.M. of x, y and z. 7. To find the least number which when divided by x, y and z leaves the remainders a, b and c, respectively. It is always observed that (x – a) = (y – b) = (z – c) = k (say) Required number = (L.C.M. of x, y and z) – k. 8. To find the least number which when divided by x, y and z leaves the same remainder r in each case. Required number = (L.C.M. of x, y and z) + r. 9. To find the greatest number that will divide x, y and z leaving the same remainder in each case. (a) When the value of remainder r is given: Required number = H.C.F. of (x – r), (y – r) and (z – r). (b) When the value of remainder is not given: Required number = H.C.F. of (x – y), (y – z) and (z – x) 10. To find the n-digit greatest number which, when divided by x, y and z. (a) leaves no remainder (i.e. exactly divisible) Step 1 L.C.M. of x, y and z = L L ) n – digit greatest number ( Step 2 remainder = R Step 3 Required number = n-digit greatest number – R (b) leaves remainder K in each case Required number = (n-digit greatest number – R) + K. 11. To find the n-digit smallest number which when divided by x, y and z (a) leaves no remainder (i.e. exactly divisible) Step 1 L.C.M. of x, y and z = L L )n-digit smallest number( Step 2 remainder = R Step 3 Required number = n-digit smallest number + (L – R). (b) leaves remainder K in each case. Required number = n-digit smallest number + (L – R) + k. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 9 EXERCISE 1. The product of 4 consecutive positive integers greater than 1 is always divisible by (1) 24 (2) 48 (3) 72 (4) 120 2. Which of the following sets of numbers are relative primes? (1) 51, 85 (2) 26, 65 (3) 57, 76 (4) 29, 75 (1) Only (2) (2) Only (4) (3) Both (1) and (3) (4) All the above 3. What is the cube root of 2 8 3 27 54? (1) 2 2  3 2  3 (2) 2 4 3 15 (3) 2 3 3 10 (4) 6 30 4. What is the cube root of 2 2 3 2 4 2 6 2 8 2 9 2 6? (1) 2 9  6 3 (2) 2 11  3 2 (3) 2 5  3 3 (4) 2 9  3 3 5. How many of the following numbers 21 2 ,25 3 ,36 2 ,18 2 are divisible by 3 3 ? (1) 1 (2) 2 (3) 3 (4) 4 6. 1 3 +2 3 +3 3 +4 3 +5 3 = (1) 20 2 (2) 15 2 (3) 25 2 (4) 12 2 7. How many digits are required for numbering the pages of a book containing 1000 pages? (1) 1000 (2) 2892 (3) 3000 (4) 3126 8. What is the total number of divisors of 1200? (1) 15 (2) 14 (3) 30 (4) 60 9. What is the total number of divisors of 5040, including one and itself? From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 10 (1) 54 (2) 60 (3) 27 (4) 25 10. A number when divided by 391 gives a remainder of 49. Find the remainder when it is divided by 13. (1) 10 (2) 9 (3) 8 (4) Cannot be determined 11. The least number that should be added or subtracted from 13218 to make it a perfect square is (1) 7 should be added. (2) 4 should be subtracted (3) 8 should be added (4) 3 should be subtracted 12. Find the value of K, if the number 1233K5 is divisible by 125. (1) 7 (2) 2 (3) 3 (4) 5 13. If the number 24P890 is divisible by 9, find the value of P. (1) 1 (2) 4 (3) 3 (4) 2 14. Evaluate : (7 5 -7 4 )/14 (1) 6  7 3 (2) 3  7 4 (3) 3  7 3 (4) 3  7 4 15. Find the sum of divisors of 480. (1) 1516 (2) 1512 (3) 1526 (4) 1412 16. What is the units digit of 653 51 ? (1) 1 (2) 3 (3) 7 (4) 9 17. What is the digit in the units place of the product 23 49  51 36 ? (1) 1 (2) 3 (3) 7 (4) 4.9 18. How many number between 100 and 300 are divisible by 11 (1) 11 From the desk of SOHAIL MERCHANT ______________________________________________________________________ - 11 - (2) 10 (3) 12 (4) 18 19. The units digit in the sum 364 102 + 364 101 is (1) 4 (2) 6 (3) 0 (4) 8 20. Which of the following is a multiple of 8? (1) 468210 (2) 469828 (3) 4692304 (4) 4695028 21. The sum of first ‗r‘ even numbers is (1) r 2 (2) r (r+1) (3) r 2 + 2r (4) r(r-1) 22. The number 2837393449 is divisible by (1) 5 (2) 7 (3) 9 (4) 11 23. Which of the following numbers is exactly divisible by 11? (1) 27184 (2) 68039 (3) 587247 (4) 92939 24. The least number which when divided by 2, 3, 4, 5 or 6 leaves a remainder of 1 in each case is (1) 162 (2) 121 (3) 221 (4) 61 25. Find the least natural number which when divided by 18,24 and 30 leaves remainders 14, 20, and 26 respectively. (1) 256 (2) 356 (3) 456 (4) 326 26. Which of the following numbers, when divided by 10 leaves a remainder of 5, when divided by 20 leaves a remainder of 15, and when divided by 30 leaves a remainder of 25? (1) 135 (2) 165 (3) 115 (4) 65 27. Find the greatest number that will divide 55, 127 and 175 leaving the same remainder in each case. (1) 24 (2) 16 From the desk of SOHAIL MERCHANT ______________________________________________________________________ - 12 - (3) 18 (4) 15 28. Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3.85 m, and 12.95 m. (1) 30 cm (2) 35 cm (3) 45 cm (4) 38 cm 29. Find the greatest number of 4 digits and the least number of 5 digits, such that they have 144 as their HCF. (1) 9999, 10000 (2) 9936, 10080 (3) 9946, 10070 (4) 9956, 10090 30. Find the least positive whole number that should be added to 1515 to make it a perfect square. (1) 3 (2) 4 (3) 6 (4) 5 31. How many numbers are there in between 1 and 50 that are divisible by 3 or 4? (1) 28 (2) 24 (3) 30 (4) 32 32. A person distributes chocolates to some children. If the gives 3 chocolates to each child, he is left with 1 chocolate. If he give 4 chocolates to each child, still he is left with 1 chocolate. But if he gives 5 chocolates to each child, he is left with none. Find the least possible number of chocolates. (1) 13 (2) 26 (3) 25 (4) 11 33. 1996 question papers are to be packed in bundles so that the number of question papers in each bundle should be equal to the total number of bundle. If the objective is to pack a maximum possible number of question papers in a bundle, how many question papers have to be left out from packing? (1) 60 (2) 90 (3) 70 (4) 80 34. Three men start together to walk along a road at the same rate. The length of their strides are respectively 68 cms, 51 cms and 85cms. How far will they go before they are ―in-step‖ again? (1) 102 m (2) 1020 m (3) 10.2m (4) 150 m 35. A supervisor was employed on the condition that he will paid highest wages per day. The money to be paid was Rs.1184 as per contract. But was finally paid Rs.1073. For how many days did he actually work? (1) 29 (2) 35 (3) 37 (4) 31 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 13 RATIOS, PROPORTIONS AND VARIATION A ratio is a comparison of two quantities by division. It is a relation that one quantity bears to another with respect to magnitude. In other words, ratio means what part one quantity is of another. The quantities may be of same kind or different kinds. For example, when we consider the ratio of the weight 45 kg of a bag of rice to the weight 29kg of a bag of sugar we are considering the quantities of same kind but when we talk of allotting 2 cricket bats to 5 sportsmen, we are considering quantities of different kinds. Normally, we consider the ratio between quantities of the same kind. If a and b are two numbers, the ratio of a to b is a/b or a +b and is denoted by a : b. The two quantities that are being compared are called terms. The first is called antecedent and the second term is called consequent. For example, the ratio 3 : 5 represents 3/5 with antecedent 3 and consequent 5. Note: 1. A ratio is a number, so to find the ratio of two quantities, they must be expressed in the same units. 2. A ratio does not change if both of is terms are multiplied or divided by the same number. Thus, 2/3= 4/6 = 6/9 etc. TYPES OF RATIOS 1. Duplicate Ratio The ratio of the squares of two numbers is called the duplicate ratio of the two numbers. For example, 3 2 /4 2 or 9/16 is called the duplicate ratio of ¾. 2. Triplicate Ratio The ratio of the cubes of two numbers is called the triplicate ratio of the two numbers. For example, 3 3 /4 3 or 27/64 is triplicate ratio of ¾. 3. Sub-duplicate Ratio The ratio of the square roots of two numbers is called the sub-duplicate ratio of two numbers. For example, 3/4 is the sub- duplicate ratio of 9/16. 4. Sub-duplicate Ratio The ratio of the cube roots of two numbers is called the sub-triplicate ratio of two numbers. For example, 2/3 is the sub-triplicate ratio of 8/27. 5. Inverse Ratio or Reciprocal Ratio If the antecedent and consequent of a ratio interchange their places, the new ratio is called the inverse ratio of the first. Thus, if a : b be the given ratio, then 1/a : 1/b or b : a is its inverse ratio. For example, 3/5 is the inverse ratio of 5/3. 6. Compound Ratio The ratio of the product of the antecedents to that of the consequents of two or more given ratios is called the compound ratio. Thus, if a :b and c:d are two given rations, then ac : bd is the compound ratio of the given ratios, For example, if ¾, 4/5 and 5/7 be the given ratios, then their compound ratios is 3×4×5/ 4×5×7, that is, 3/7. PROPORTION The equality of two ratios is called proportion. If a/b = c/d, then a, b, c and d are said to be in proportion and we write a : b: : c: d. This is read as ―a is to b as c is to d‖. For example, since ¾ = 6/8, we write 3; 4: : 6: 8 and say 3, 4, 6 and 8 are in proportion. Each term of the ratio a/b and c/d is called a proportional. a, b, c, and d are respectively the first, second, third and fourth proportionals Here, a, d are known as extremes and b, c are known as means. SOME BASIC FORMULAE 1. If four quantities are in proportion, then product of Means = product of Extremes For example, in the proportion a : b: : c: d, we have bc = ad. From this relation we see that if any three of the four quantities are given, the fourth can be determined. 2. Fourth proportional If a: b: :c :x, x is called the fourth proportional of a, b, c. We have, a/b = c/x or, x = b×c/a Thus, fourth proportional of a, b, c is b × c / a. Illustrational 1 Find a fourth proportional to the numbers 2, 5, 4. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 14 Solution Let x be the fourth proportional, then 2 : 5 : : 4 : x or 2/5 = 4/x. x = 5 × 4 /2 = 10. 3. Third proportional If a: b: : c: x, x is called the third proportional of a, b. We have, a/b= b/x or x= b 2 /a. Thus, third proportional of a. b is b 2 /a Illustration 2 Find a third proportional to the numbers 2.5, 1.5 Solution Let x be the third proportional, then 2.5 : 1.5 : :1.5 : x or 2.5/1.5= 1.5/x. x = 1.5 × 1.5/2.5 = 0.9 4. Mean Proportional If a: x: : x: b, x is called the mean or second proportional of a, b. We have, a/x =x/b or x 2 = ab or x = \ab Mean proportional of a and b is \ab. We also say that a, x, b are in continued proportion Illustration 3 Find the mean proportional between 48 and 12. Solution Let x be the mean proportional. Then, 48 : x : : x : 12 or, 48/x = x/12 or, x 2 = 576 or, x=24. 5. If a/b = c/d, then (i) (a + b)/b = (c +d)/d (Componendo) (ii) (a – b)/b = (c-d)/d (Dividendo) (iii) (a + b)/a-b = c +d/c-d = (Componendo and Dividendo) (iv) a/b = a + c/b+d = (a –c)/b-d Illustration 4 The sum of two umber is c and their quotient is p/q. Find the numbers. Solution Let the numbers be x, y. Given: x + y = c …(1) and, x/y = p/q …(2) x/ x+y = p/p+q ¬ x/c = p/p+q [Using (1)] ¬ x = pc/p +q. SOME USEFUL SHORT-CUT METHODS 1. (a) If two numbers are in the ratio of a: b and the sum of these numbers is x, then these numbers will be ax/ a + b and bx/ a+b, respectively. or If in a mixture of x liters of, two liquids A and B in the ratio of a: b, then the quantities of liquids A and B in the mixture will be ax / a + b litres and bx/ a + b litres, respectively. (b) If three numbers are in the ratio a : b: c and the sum of these numbers is x, then these numbers will be ax / a + b + c , bx / a + b + c and cx / a + b + c, respectively. Explanation Len the three numbers in the ratio a: b: c be A, B and C. Then, A = ka, B = kb, C =kc and, A + B + C = ka + kb + kc = x ¬ k(a+b+c) = x ¬ k = x / a + b+ c. A = ka = ax / a + b+ c. B = kb = bx / a + b+ c. C = kc = cx / a + b+ c. Illustration 5 Two numbers are in the ratio of 4 : 5 and the sum of these numbers is 27. Find the two numbers. Solution Here, a = 4, b = 5, and x = 27. The first number = ax / a + b = 4 × 27 / 4+5 = 12. and, the second number = bx / a + b = 5 × 27 / 4+5 = 15. Illustration 6 Three numbers are in the ratio of 3: 4 : 8: and the sum of these numbers is 975. Find the three numbers. Solution Here, a = 3, b = 4, c = 8 and x = 975 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 15 The first number = ax / a + b+ c = (3 ×975)/ 3 + 4 + 8 = 195. The second number = bx / a + b+ c = (4 ×975)/ 3 + 4 + 8 = 260. and, the third number = cx / a + b+ c = (8 ×975)/ 3 + 4 + 8 = 520. 2. If two numbers are in the ratio of a : b and difference between these is x, then these numbers will be a) ax/ a-b and bx/ a-b, respectively (where a > b). b) ax/ a-b and bx/ a-b, respectively (where a < b). Explanation Let the two numbers be ak and bk. Let a > b. Given : ak – bk = x ¬ (a – b)k = x or k = x / (a-b). Therefore, the two numbers are ax / a-b and bx/ a-b. Illustration 7 Two numbers are in the ratio of 4 : 5. If the difference between these numbers is 24, then find the numbers. Solution Here, a = 4, b = 5 and x = 24. The first number = ax/ b-a = 4 ×24/5- 4 = 96 and, the second number = bx/ b-a = 5× 24 / 5-4 = 120. 3. (a). If a : b = n 1 : d 1 and b : c = n 2 : d 2 , then a : b : c = (n 1 ×n 2 ) : (d 1 × n 2 ) : (d 1 × d 2 ). (b). If a : b = n 1 : d 1 , b : c = n 2 : d 2 , and c : d = n 3 : d 3 then a : b : c : d= (n 1 × n 2 × n 3 ) : (d 1 × n 2 × n 3 ) : (d 1 × d 2 × n 3 ) : (d 1 × d 2 × d 3 ). Illustration 8 If A : B = 3 : 4 and B : C = 8 : 9, find A : B : C. Solution Here, n 1 = 3, n 2 =8, d 1 =4 and d 2 = 9. a : b : c = (n 1 ×n 2 ) : (d 1 ×n 2 ) : (d 1 ×d 2 ) = (3×8) : (4×8) : (4×9) = 24 : 32 : 36 or, 6: 8 : 9. Illustration 9 If A : B = 2 : 3, B:C = 4 : 5 and C : D = 6 : 7, find A :D. Solution Here, n 1 = 2, n 2 = 4, n 3 = 6, d 1 = 3, d 2 = 5 and d 3 = 7. A : B : C : D = (n 1 ×n 2 ×n 3 ) : (d 1 ×n 2 ×n 3 ) : (d 1 × d 2 × n 3 ) : (d 1 × d 2 × d 3 ) = (2 × 4 × 6) : (3 × 4 × 6) : (3 × 5 × 6) : (3 ×5 ×7) = 48 : 72 : 90: 105: or, 16: 24 : 30 ; 35. Thus, A : D = 16 : 35. 4. (a) The ratio between two numbers is a : b. If x is added to each of these numbers, the ratio becomes c : d. The two numbers are given as: ax(c – d) / ad – bc and bx(c- d) / ad –bc. Explanation Let two number be ak and bk. Given : ak +x / bk+x = c/d ¬akd +dx= cbk + cx ¬ k(ad –bc) = x(c –d) ¬k =x(c-d)/ ad – bc. Therefore, the two numbers are ax(c-d) / ad-bc and bx(c-d) / ad- bc (b) The ratio between two numbers is a : b. if x is subtracted from each of these numbers, the ratio becomes c : d. The two numbers are given as: ax(d-c) / ad-bc and bx(d-c) / ad- bc Explanation Let the two numbers be ak and bk. Given : ak-x/bk-x = c/d ¬ akd-xd = bck-xc ¬ k(ad-bc) = x(d-c) ¬ k = x(d-c)/ad-bc. Therefore, the two numbers are ax(d-c)/ad-bc and bx(d-c)/ad-bc Illustration 10 Given two numbers which are in the ratio of 3 : 4, If 8 is added to each of them, their ratio is changed to 5 : 6. Find two numbers. Solution We have, From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 16 a : b = 3 : 4, c : d = 5 : 6 and x = 8. The first number = ax(c – d)/ ad –bc = 3 × 8×(5-6) / (3× 6- 4 ×5) = 12. and, the second number = bx(c – d)/ ad –bc = 4 × 8×(5-6) / (3× 6- 4 ×5) = 16. Illustration 11 The ratio of two numbers is 5 : 9. If each number is decreased by 5, the ratio becomes 5: 11. Find the numbers. Solution We have, a : b = 5: 9, c: d = 5: 11 and x =5. The first number = ax (d – c)/ ad –bc = 5 × 5×(11-5)/ (5×11- 9 ×5) = 15. and, the second number= bx(d-c)/ad-bc 9×5×(11-5)/(5×11-9×5)= 27. 5. (a) If the ratio of two numbers is a: b, then the numbers that should be added to each of the numbers in order to make this ratio c : d is given by ad-bc/ c-d. Explanation Let the required number be x. Given: a+x/ b+x = c/d ¬ ad+ xd = bc + xc ¬ x(d-c) =bc –ad or, x =ad-bc/c-d. (b)If the ratio of two numbers is a : b, then the number that should be subtracted from each of the numbers in order to make this ratio c : d is given by bc-ad/c-d. Explanation Let the required number be x. Given: a-x/ b-x = c/d ¬ ad- xd = bc - xc ¬ x(c-d) =bc –ad or, x = bc-ad/c-d. Illustration 12 Find the number that must be subtracted from the terms of the ratio 5 : 6 to make it equal to 2 : 3. Solution We have a : b= 5 : 6 and c: d =2 : 3. The required number = bc-ad/ c-d = 6 × 2-5×3/ 2-3 =3. Illustration 13 Find the number that must be added to the terms of the ratio 11 : 29 to make it equal to 11 : 20. Solution We have, a : b= 11 : 29 and c: d =11: 20. The required number = ad-bc/ c-d = 11 × 20-29×11/ 11-20 = 11. EXERCISE 1. Divide Rs.1870 into three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are equal. 1. 241, 343, 245 2. 400, 800, 670 3. 470, 640, 1160 4. None 2. Divide Rs.500 among A, B, C and D so that A and B together get thrice as much as C and D together, B gets four times of what C gets and C gets 1.5 times as much as D. Now the amount c gets? 1. 300 2. 75 3. 125 4. None 3. If 4 examiners can examine a certain number of answer books in 8 days by working 5 hours a day, for how many hours a day would 2 examiners have to work in order to examine twice the number of answer books in 20 days. 1. 6 2. 1/2 3. 8 4. 9 4. In a mixture of 40 liters, the ratio of milk and water is 4:1. How much water much be added to this mixture so that the ratio of milk and water becomes 2:3 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 17 1. 20 litres 2. 32 litres 3. 40 litres 4. 30 litres 5. If three numbers are in the ratio of 1:2:3 and half the sum is 18, then the ratio of squares of the numbers is: 1. 6:12:13 2. 1:2:4 3. 36:144:324 4. None 6. The ratio between two numbers is 3:4 and their LCM is 180. the first number is: 1. 60 2. 45 3. 15 4. 20 7. A and B are tow alloys of argentums and brass prepared by mixing metals in proportions 7:2 and 7:11 respectively. If equal quantities of the two alloys are melted to form a third alloy C, the proportion of argentums and brass in C will be: 1. 5:9 2. 5:7 3. 7:5 4. 9:5 8. If 30 men working 7 hours a day can do a piece of work in 18 days, in how many days will 21 men working 8 hours a day do the same work? 1. 24 days 2. 22.5 days 3. 30 days 4. 45 days 9. The incomes of A and B are in the ratio 3:2 and their expenditure are in the ratio 5:3. If each saves Rs.1000, then, A‘s income is 1. 3000/- 2. 4000/- 3. 6000/- 4. 9000/- 10. If the ratio of sines of angles of a triangle is 1:1:\2, then the ratio of square of the greatest side to sum of the squares of other two sides is 1. 3:4 2. 2:1 3. 1:1 4. Cannot be determined 11. Divide Rs.680 among A, B and C such that A gets 2/3 of what B gets and B gets 1/4 th of what C gets. Now the share of C is? 1. 480/- 2. 300/- 3. 420/- 4. None 12. A, B, C enter into a partnership. A contributes one-third of the whole capital while B contributes as much as A and C together contribute. If the profit at the end of the year is Rs.84, 000, how much would each received? 1. 24,000, 20,000, 40,000 2. 28,000, 42,000, 14,000 3. 28,000, 42,000, 10,000 4. 28,000, 14,000, 42,000 13. The students in three batches at AMS Careers are in the ratio 2:3:5. If 20 students are increased in each batch, the ratio changes to 4:5:7. the total number of students in the three batches before the increases were 1. 10 2. 90 3. 100 4. 150 14. The speeds of three cars are in the ratio 2:3:4. The ratio between the times taken by these cars to travel the same distance is 1. 2:3:4 2. 4:3:2 3. 4:3:6 4. 6:4:3 15. Rs.2250 is divided among three friends Amar, Bijoy and Chandra in such a way that 1/6 th of Amar‘s share, 1/4 th of Bijoy‘s share and 2/5 th of chandra‘s share are equal. Find Amar‘s share. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 18 1. 720/- 2.1080/- 3. 450/- 4. 1240/- 16. After an increment of 7 in both the numerator and denominator, a fraction changes to ¾. Find the original fraction. 1. 5/12 2. 7/9 3. 2/5 4. 3/8 17. The difference between two positive numbers is 10 and the ratio between them is 5:3. Find the product of the two numbers. 1. 375 2. 175 3. 275 4. 125 18. If 30 oxen can plough 1/7 th of a field in 2 days, how many days 18 oxen will take to do the remaining work? 1. 30 days 2. 20 days 3. 15 days 4. 18 days 19. A cat takes 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog? 1. 11:15 2. 15:11 3. 16:15 4. 15:16 20. The present ratio of ages of A and B is 4:5. 18 years ago, this ratio was 11:16. Find the sum total of their present ages. 1. 90 years 2. 105 years 3. 110 years 4. 80 years 21. Three men rent a farm for Rs.7000 per annum. A puts 110 cows in the farm for 3 months, B puts 110 cows for 6 months and C puts 440 cows for 3 months. What percentage of the total expenditure should A pay? 1. 20% 2. 14.28% 3. 16.66% 4. 11.01% 22. 10 students can do a job in 8 days, but on the starting day, two of them informed that they are not coming. By what fraction will the number of day required for doing the whole work get increased? 1. 4/5 2. 3/8 3. 3/4 4. 1/4 23. A dishonest milkman mixed 1 liter of water for every 3 liters of milk and thus make up 36 liters of milk. If he now adds 15 liters of milk to the mixture, find the ratio of milk and water in the new mixture. 1. 12:5 2. 14:3 3. 7:2 4. 9:4 24. Rs.3000 is distributed among A, B and C such that A gets 2/3 rd of what B and C together get and C gets ½ of what A and B together get. Find C‘s share 1. 750/- 2. 1000/- 3. 800/- 4. 1200/- 25. If the ratio of the ages of Maya and Chhaya is 6:5 at present, and fifteen years from now, the ratio will get changed to 9:8, then find Maya‘s present age. 1. 24 years 2. 30 years 3. 18 years 4. 33 years 26. If Rs.58 is divided among 150 children such that each girl and each boy gets 25 p and 50 p respectively. Then how many girls are there? 1. 52 2. 54 3. 68 4. 62 27. If 391 bananas were distributed among three monkeys in the ratio ½:2/3:3/4, how many bananas did the first monkey get? 1. 102 2. 108 3. 112 4. 104 5 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 19 28. A mixture contains milk and water in the ratio 5:1. On adding 5 liters of water, the ratio of milk to water becomes 5:2. the quantity of milk in the mixture is: 1. 16 litres 2. 25 litres 3. 32.5 litres 4. 22.75 litres 29. A beggar had ten paise, twenty paise and one rupee coins in the ratio 10:17:7 respectively at the end of day. If that day he earned a total of Rs.57, how many twenty paise coins did he have? 1. 114 2. 171 3. 95 4. 85 30. Vijay has coins of he denomination of Re.1, 50 p and 25 p in the ratio of 12:10:7. The total worth of the coins he has is Rs.75. Find the number of 25 p coins that Vijay has 1. 48 2. 72 3. 60 4. None 31. If two numbers are in the ratio of 5:8 and if 9 be added to each, the ratio becomes 8:11. Now find the lower number. 1. 5 2. 10 3. 15 4. None 32. A cask contains a mixture of 49 liters of wine and water in the proportion 5:2. How much water must be added to it so that the ratio of wine to water may be 7:4? 1. 3, 5 2. 6 3. 7 4. None 33. A cask contains 12 gallons of mixture of wine and water in the ratio 3:1. How much of the mixture must be drawn off and water substituted, so that wine and water in the cask may become half and half. 1. 3 litres 2. 5 litres 3. 6 litres 4. None of these 34. The total number of pupils in three classes of a school is 333. the number of pupils in classes I and II are in the ratio 3:5 and those in classes II and III are in the ratio 7:11. Find the number of pupils in the class that had the highest number of pupils. 1. 63 2. 105 3. 165 4. 180 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 20 PERCENTAGES Introduction The term per cent means per hundreds or for every hundred. It is the abbreviation of the Latin phrase per centum. Scoring 60 per cent marks means out of every 100 marks the candidate scored 60 marks. The term per cent is sometimes abbreviated as p.c. The symbol % is often used for the term per cent. Thus, 40 per cent will be written as 40%. A fraction whose denominator is 100 is called a percentage and the numerator of the fraction is called rate per cent, e.g. 5/100 and 5 per cent means the same thing, i.e. 5 parts out of every hundred parts. 1. To Convert a fraction into a per cent: to convert any fraction l/m to rate per cent, multiply it by 100 and put % sign, i.e. l/m × 100% 2. To Convert a Percent into a Fraction: To convert a per cent into a fraction , drop the per cent sign and divide the number by 100. 3. To find a percentage of a given number: x % of given number (N) = x/100 × N. Some useful shortcut methods 1. (a) if A is x% more than that of B, then B is less than that of A by % 100 100 ( ¸ ( ¸ × + x x (b) If a is x% less than that of B, then B is more than that of A by % 100 100 ( ¸ ( ¸ × ÷ x x 2. If a is x% of C and B is y% of C, then A = x/y × 100% of B. 3. (a) If two numbers are respectively x% and y% more than a third number, then the first number is % 100 100 100 | | . | \ | × + + y x of the second and the second is % 100 100 100 | . | \ | × + + x y of the first. (b) If two numbers are respectively x% and y% less than a third number, then the first number is % 100 100 100 | | . | \ | × ÷ ÷ y x of the first. 4. (a) If the price of a commodity increases by P%, then the reduction in consumption so as not to increase the expenditure is % 100 100 | . | \ | × +P P . (b) If the price of a commodity decreases by p%, then the increase in consumption so as not to decrease the expenditure is % 100 100 | . | \ | × ÷P P . 5. If a number is changed (increased/decreased) successively by x% and y%, then net% change is given by (x+y+(xy/100))% which represents increase or decrease in value according as the sign is +ve or –ve. If x or y indicates decrease in percentage, then put –ve sign before x or y, otherwise +ve sign. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 21 6. If two parameters A and B are multiplied to get a product and if A is changed (increased/decreased) by x% and another parameter B is changed (increased/decreased) by y%, then the net% change in the product (A × B) is given (x+y+(xy/100))% which represents increase or decrease in value according as the sign in +ve or –ve. If x or y indicates decrease in percentage, then put –ve sign before x or y, otherwise +ve sign. 7. If the present population of a town (or value of an item) be P and the population (or value of item) changes at r% per annum, then (a) Population (or value of item) after n years = n r P | . | \ | + 100 1 (b) Population (or value of item) n years ago = n r P | . | \ | + 100 1 where r is +ve or –ve according as the population (or value of item) increase or decreases. 8. If a number A is increased successively by x% followed by y% and then by z%, then the final value of A will be | . | \ | + | . | \ | + | . | \ | + 100 1 100 1 100 1 z y x A In case a given value decreases by any percentage, we will use a negative sign before that. 9. In an examination, the minimum pass percentage is x%. If a student secures y marks and fails by z marks, then the maximum marks in the examination is x z y ) ( 100 + . 10. In an examination x% and y% students respectively fail in two different subjects while z% students fail in both the subjects, then the percentage of students who pass in both the subjects will be (100-(x+y-z))%. EXERCISE 1. What is 20% of 50% of 75% of 70? 1. 5.25 2. 6.75 3. 7.25 4. 5.50 2. Ram sells his goods 25% cheaper than Shyam and 25% dearer than Bram. How much percentage is Bram‘s goods cheaper than Shyam‘s? 1. 33.33% . 50% 3. 66.66% 4. 40% 3. In an election between 2 candidates, Bhiku gets 65% of the total valid votes. If the total votes were 6000. What is the number of valid votes that the other candidate Mhatre gets if 25% of the total votes were declared invalid? 1. 1625 2. 1575 3. 1675 4. 1525 4. In a medical certificate, by mistake a candidate gave his height as 25% more than normal. In the interview panel, he clarified that his height was 5 feet 5 inches. Find the percentage correction made by the candidate from his stated height to his actual height. 1. 20 2. 28.56 . 25 4. None 5. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 22 6. Arjit Sharma generally wears his father‘s coat. Unfortunately, his cousin Shaurya poked him one day that he was wearing a coat of length more than his height by 15%. If the length of Arjit‘s father‘s coat is 120 cm then find the actual length of his coat. 1. 105 2. 108 3. 104.34 4. 102.72 7. ***In a mixture of 80 liters of milk and water, 25% of the mixture is milk. How much water should be added to the mixture so that milk becomes 20% of the mixture? 1. 20 liters 2. 15 liters 3. 25 liters 4. None 8. 50% of a% of b is 75% of b% of c. Which of the following is c? 1. 1.5a 2. 0.667a 3. 0.5a 4. 1.25a 9. ***A landowner increased the length and the breadth of a rectangular plot by 10% and 20% respectively. Find the percentage change in the cost of the plot assuming land prices are uniform throughout his plot. 1. 33% 2. 35% 3. 22.22% 4. None 10. The height of a triangle is increased by 40%. What can be the maximum percentage increase in length of the base so that the increase in area is restricted to a maximum of 60%? 1. 50% 2. 20% 3. 14.28% 4. 25% 11. The length, breadth and height of a room in the shape of a cuboid are increased by 10%, 20% and 50% respectively. Find the percentage change in the volume of the cuboids. 1. 77% 2. 75% 3. 88% 4. 98% 12. The salary of Amit is 30% more than that of Varun. Find by what percentage is the salary of Varun less than that of Amit? 1. 26.12% 2. 23.07% 3. 21.23% 4. None 13. ***The price of sugar is reduced by 25% but in spite of the decrease, Aayush ends up increasing his expenditure on sugar by 20%. What is the percentage change in his monthly consumption of sugar? 1. +60% 2. –10% 3. +33.33% 4. 50% 14. The price of rice falls by 20%. How much rice can be bought now with the money that was sufficient to buy 20 kg of rice previously? 1. 5kg 2. 15 kg 3. 25 kg 4. 30 kg 15. 30% of a number when subtracted from 91, gives the number itself. Find the number. 1. 60 2. 65 3. 70 4. None 16. ***At an election, the candidate who got 56% of the votes cast won by 144 votes. Find the total number of voters on the voting list if 80% people cast their vote and there were no invalid votes. 1. 360 2. 720 3. 1800 4. 1500 17. The population of a village is 1,00,000. The rate of increase is 10% per annum. Find the population at the start of the third year? 1. 1, 33,100 2. 1, 21, 000 3. 1, 20, 000 4. None From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 23 18. the population of the village of Gavas Is 10, 000 at this moment. It increases by 10% in the first year. However, in the second year, due to immigration, the population drops by 5%. Find the population at the end of the third year if in the third year the population increases by 20%. 1. 12, 340 2. 12, 540 3. 1, 27, 540 4. 12, 340 19. A man invests Rs.10,000 in some shares in the ratio 2:3:5 which pay dividends of 10%, 25% and 20% (on his investment) for that year respectively. Find his dividend income. 1. 1900 2. 2000 3. 2050 4. 1950 20. *In an examination, Mohit obtained 20% more than Sushant but 10% less than Rajesh. If the marks obtained by Sushant is 1080, find the percentage marks obtained by Rajesh if the full marks is 2000. 1. 86.66% 2. 72% 3. 78.33% 4. None 21. In a class, 25% of the students were absent for an exam. 30% failed by 20 marks and 10% just passed because of grace marks of 5. Find the average score of the class if the remaining students scored an average of 60 marks and the pass marks are 33 (counting the final scores of the candidates). 1. 37.266 2. 37.6 3. 37.8 4. 36.93 22. Ram spends 20% of his monthly income on his household expenditure. 15% of the rest on books, 30% of the rest on clothes and saves the rest. On counting, he comes to know that he has finally saved Rs.9520. Find his monthly income. 1. 10000 2. 15000 3. 20000 4. None 23. Hans and Bhaskar have salaries that jointly amount of Rs.10,000 per month. They spend the same amount monthly and then it is found that the ratio of their savings is 6:1. Which of the following is Hans‘s salary? 1. 6000/- 2. 5000/- 3. 4000/- 4. 3000/- 24. The population of a village is 5500. If the number of males increases by 11% and the number of females increases by 20%, then the population becomes 6330. Find the population of females in the town. 1. 2500 2. 3000 3. 2000 4. 3500 25. Vicky‘s salary is 75% more than Ashu‘s. Vicky got a raise of 40% on his salary while Ashu got a raise of 25% on his salary. By what percent is Vicky‘s salary more than Ashu‘s? 1. 96% 2. 51.1% 3. 90% 4. 51.1% 26. On a shelf, the first row contains 25% more books than the second row and the third row contains 25% less books then the second row. If the total number of books contained in all the rows is 600, then find the number of books in the first row. 1. 250 2. 225 3. 300 4. None 27. An ore contains 25% of an alloy that has 90% iron. Other than this, in the remaining 75% of the ore, there is no iron. How many kilograms of the ore are needed to obtain 60 kg of pure iron? 1. 250kg 2. 275 kg 3. 300 kg 4. 266.66 kg From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 24 28. Last year, the Indian cricket team played 40 one-day cricket matches out of which they managed to win only 40%. This year, so far it has played some matches, which has made it mandatory for it to win 80% of the remaining matches to maintain its existing winning percentage. Find the number of matches played by India so far this year? 1. 30 2. 25 3. 28 4. Insufficient information 29. In the recent, climate conference in New York, out of 700 men, 500 women, 800 children present inside the building premises, 20% of the men, 40% of the women and 10% of the children were Indians. Find the percentage of people who were not Indian? 1. 73% 2. 77% 3. 79% 4. 83% 30. Ram sells his goods 20% cheaper than Bobby and 20% dearer than Chandilya. How much percentage is Chandilya‘s goods cheaper/dearer than Bobby‘s 1. 33.33% 2. 50% 3. 42.85% 4. None 31. Out of the total production of iron from hematite, an ore of iron, 20% of the ore gets wasted, and out of the remaining iron, only 25% is pure iron. If the pure iron obtained in a year from a mine of hematite was 80, 000 kg, then the quantity of hematite mined from that mine in the year is 1. 5, 00, 000 kg 2. 4, 00, 000 kg 3. 4, 50, 000 kg 4. None 32. Recently, while shopping in Patna Market in Bihar, I came across two new shirts selling at a discount. I decided to buy one of them for my little boy Sherry. The shopkeeper offered me the first shirt for Rs.42 and said that it usually sold for 8/7 of that price. He then offered me the other shirt for Rs.36 and said that it usually sold for 7/6 th of that price. Of the two shirts which one do you think is a better bargain and what is the percentage discount on it? 1. First shirt, 12.5% 2. second shirt, 14.28% 3. Both are same 4. None 33. 4/5 th of the voters in Bellary promised to vote for Sonia Gandhi and the rest promised to vote for Sushma Swaraj. Of these voters, 10% of the voters who had promised to vote for Sonia Gandhi did not vote on the Election Day, while 20% of the voters who had promised to vote for Sushma Swaraj did not vote on the Election Day. What is the total no. of votes polled if Sonia Gandhi got 216 votes? 1. 200 2. 300 3. 264 4. 100 34. Ravana spends 30% of his salary on house rent, 30% of the rest he spends on his children‘s education and 24% of the total salary he spends on clothes. After his expenditure, he is left with Rs.2500. What is Ravana‘s salary? 1. 11, 494, 25/- 2. 20, 000/- 3. 10, 000/- 4. 15, 000/- 35. The entrance ticket at the Minerva theatre in Mumbai is worth Rs.250. When the price of the ticket was lowered, the sale of tickets increased by 50% while the collections recorded a decrease of 17.5%. Find the deduction in the ticket price. 1. 150/- 2. 112.5/- 3. 105/- 4. 120/- 36. In the year 2000, the luxury car industry had two car manufacturers—Maruti and Honda with market shares of 25% and 75% respectively. In 2001, the overall market for the product increased by 50% and a new player BMW also entered the market and captured 15% of the market share. If we know that the market share Maruti increased to 50% in the second year, the share of Honda in that year was: 1. 50% 2. 45% 3. 40% 4. 35% From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 25 Averages & Mixtures Whenever we are asked the marks scored by us in any examination, we usually tell the marks in percentage, taking the percentage of total marks of all subjects. This percentage is called average percentage. Also, in a class, If there are 100 students, instead of knowing the age of individual student, we usually talk about average age. The average or mean or arithmetic of a number of quantities of the same kind is equal to their sum divided by the number of those quantities. For example, the average of 3, 11, 15, 18,19, and 23 is 3 + 9 +11+ 15+ 18+ 19+ 23+ /7 = 98/7 = 14. SOME BASIC FORMULAE 1. Average = sum of quantities/ Number of quantities 2. Sum of quantities = Average × Number of quantities 3. Number of quantities = Sum of quantities/ Average Illustration 1 A man purchased 5 toys at the rate of Rs 200each, 6 toys at the rate of Rs 250each and 9 toys at the rate of Rs 300 each. Calculate the average cost of one toy. Solution Price of 5 toys = 200 × 5 = Rs 1000 Price of 6 toys = 250 × 6 = Rs 1500 Price of 9 toys = 300 × 9 = Rs 2700 Average price of 1 toy = 1000 + 1500 + 2700/ 20 = 5200/20 = Rs 260. Illustration 2 The average marks obtained by 200 students in a certain examination are 45. Find the total marks. Solution Total marks = Average marks × Number of students = 200 × 45 = 900. Illustration 3 Total temperatures for the month of September is 840 0 C, If the average temperature of that month is 28 0 C, find of how many days is the month of September. Solution Number of days in the month of September = Total temperature/ Average temperature = 840/28 = 30days. SOME USEFUL SHORT–CUT METHODS 1. Average of two or more groups taken together a) If the number of quantities in two groups be n 1 and n 2 and their average is x and y, respectively, the combined average (average of all of then put together) is n 1 x +n 2 y / n 1 + n 2 Explanation No. of quantities in fist group = n 1 Their average = x Sum = n 1 × x No. of quantities in second group = n 2 Their average = y Sum = n 2 × y No. of quantities in the combined group = n 1 +n 2 Total sum (sum of quantities of first group and second group) = n 1 x+n 2 y Combined Average = n 1 x+n 2 y./ n 1 +n 2 . b). If the average of n 1 quantities is x and the average of n 2 quantities out of them is y, the average of remaining group (rest of the quantities) is From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 26 n 2 x – n 2 y/ n 1 - n 2 . Explanation No. of quantities = n 1 Their average = x Sum = n 1 x No of quantities taken out – n 2 Their average = y Sum = n 2 y Sum of remaining quantities = n 1 x – n 2 y No. of remaining quantities = n 1 – n 2 Average of remaining group = n 1 x – n 2 y/ n 1 – n 2 Illustration 4 The average weight of 24 students of section A of a class is 58 kg whereas the average weight of 26 students of section B of the same class is 60. 5 kg. Find the average weight of all the 50 students of the class. Solution Here, n 1 = 24, n 2 = 26, x = 58 and y = 60.5. Average weight of all the 50 students = n 1 x+n 2 y/ n 1 +n 2 = 24 ×58 + 26×60.5 / 24+26 = 1392 +1573/ 50 = 2965/ 50 =59.3kg. Illustration 5 Average salary of all the 50 employees including 5 officers of a company is Rs 850. If the average salary of the officers is Rs 2500, find of the class. Solution Here, n 1 = 50, n 2 =5, x = 850and y = 2500. Average salary of the remaining staff = n 1 x-n 2 y/ n 1 -n 2 = 50×850 -5×2500 / 50-5 = 42500-12500/ 45 = 30000/ 45 = Rs 667(approx) 2. If x is the average of x 1 , x 2 , …, x n , then a) The average of x 1 + a, x 2 + a, …., x n + a is x +a. b) The average of x 1 - a, x 2 - a, …., x n - a is x -a. c) The average of ax 1 , ax 2, ….,ax n is ax, provided a ≠ 0. d) The average of x 1 / a, x 2 / a, …., x n / a isx /a, provided a ≠ 0. Illustration 6 The average value of six numbers 7, 12, 17, 24, 26 and 28 is 19. If 8 is added to each number, what will be the new average? Solution The new average = x +a. = 19+8 = 27. Illustration 7 The average value of x numbers is 5x. If x – 2 is subtracted from each given number, what will be the new average? Solution The new average =x -a. = 5x- (x-2) = 4x +2. Illustration 8 The average of 8 numbers is 21.If each of the numbers multiplied by 8, find the average of a new set of numbers. Solution The average of a new set of numbers = ax = 8× 21 = 168. 3. The average of n quantities is equal to x. If one of the given quantities whose value is p, is replaced by a new quantity having value q, the average becomes y, then q = p+n(y-x) Illustration 9 The average weight of 25 persons is increased by 2 kg when one of them whose weight is 60kg, is replaced by a new person. What is the weight of the new person? From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 27 Solution The weight of the new person = p + n(y-x) = 60 + 25(2)= 110kg 4. a). The average of n quantities is equal to x. When a quantity is removed, the average becomes y. The value of the removed quantity is n(x- y)+y. b) The average of n quantities is equal to x. When a quantity is added, the average becomes y. The value of the new quantity is n(y-x)+y Illustration10 The average are of 24 students and class teacher is16 years, If the class teacher‘s age is excluded, the average age reduces by 1 year. What is the age of the class teacher? Solution The age of class teacher = n(x- y) + y = 25(16 – 15) + 15 = 40 years. Illustration 11 The average age of 30 children in a class is 9 years. If the teacher‘s age be included, the average age becomes 10years. Find the teacher‘s age. Solution The teacher‘s age = n(y- x) + y = 30(10 – 9) +100 = 40 years. 5. a).The average of first n natural numbers is (n +1) /2 b). The average of square of natural numbers till n is (n +1)(2n+1)/6. c). The average of cubes of natural numbers till n is n(n +1) 2 /4 d). The average of odd numbers from 1 to n is (last odd number +1) / 2 e). The average of even numbers from 1 to n is (last even number + 2) / 2. Illustration 12 Find the average of first 81natural number. Solution The required average = n + 1/ 2 = 81 + 1 /2 = 41. Illustration 13 What is the average of squares of the natural numbers from 1 to 41? Solution The required average = (n+1)(2n+1)/ 6 = (41+1)(2×41+1)/ 6 = 42 × 83/ 6 = 3486/ 6 = 581 Illustration 14 Find the average of cubes of natural numbers from 1 to 27. Solution The required average = n(n +1) 2 / 4 = 27×(27+1) 2 / 4 27 × 28 × 28 / 4 = 21168 / 4 = 5292. Illustration 15 What is the average of odd numbers from 1 to 40? Solution The required average = last odd number + 1/ 2 = 39 +1/ 4 =20. Illustration 16 What is the average of even numbers from 1 to 81? Solution The required average = last even number + 2/ 2 = 80+2 = 41. 6. a).If n is odd: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the middle number. b). If n is even: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the average of the middle two numbers. c). The average of first n consecutive numbers is (n+1). d). The average of first n consecutive odd numbers is n. e). The average of squares of first n consecutive even number is2 (n+1)(2n+1) / 3. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 28 f). The average of squares of consecutive even number till n is (n+)(n+2) / 3. g). The average of squares of squares of consecutive odd numbers till n is n(n+2)/ 3. h). If the average of n consecutive numbers is m, then the difference between the smallest and the largest number is 2(n-1). Illustration 17 Find the average of 7 consecutive numbers 3, 4, 5, 6, 7, 8, 9. Solution The required average= middle number=6. Illustration 18 Find the average of consecutive odd numbers 21, 23, 25, 27, 29, 31, 33, 35. Solution The required average = average of middle two numbers = average of 27 and 29 = 27+29 / 2 = 28. Illustration 19 Find the average of first 31 consecutive even numbers. Solution The required average = (n+1) = 31+ 1= 32. Illustration 20 Find the average of first 50 consecutive odd numbers. Solution The required average = n = 50. EXERCISE 1. The average of 13 papers is 40. The average of the first 7 papers is 42 and of the last seven papers is 35. Find the marks obtained in the 7 th paper? (A) 23 (B) 38 (C) 19 (D) None of these 2. The average age of the Indian cricket team playing the Nagpur test is 30. The average age of 5 of the players is 27 and that of another set of 5 players, totally different from the first five, is 29. If it is the captain who was not included in either of these two groups, then find the age of the captain. (A) 75 (B) 55 (C) 50 (D) Cannot be determined 3. A bus goes to Ranchi from Patna at the rate of 60 km per hour. Another bus leaves Ranchi for Patna at the same times as the first bus at the rate of 70 km per hour. Find the average speed for the journeys of the two buses combined if it is known that the distance from Ranchi to Patna is 420 kilometers. (A) 64.615 kmph (B) 64.5 kmph (C) 63.823 kmph (D) 64.82 kmph 4. A train travels 8 km in the first quarter of an hour, 6 km in the second quarter and 40 km in the third quarter. Find the average speed of the train per hour over the entire journey. (A) 72 km/h (B) 18 km/h (C) 77.33 km/h From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 29 (D) 78.5 km/h 5. The average weight of 6 men is 68.5 kg. If I is known that Ram and Tram weigh 60 kg each, find the average weight of the others. (A) 72.75 kg (B) 75 kg (C) 78 kg (D) None of these 6. The average score of a class of 40 students is 52. What will be the average score of the rest of the students if the average score of 10 of the students is 61. (A) 50 (B) 47 (C) 48 (D) 49 7. The average age of 80 students of IIM, Bangalore of the 1995 batch is 22 years. What will be the new average if we include the 20 faculty members whose average age is 37 years? (A) 32 years (B) 24 years (C) 25 years (D) None of these 8. Out of the three numbers, the first is twice the second and three times the third. The average of the three numbers is 88. The smallest number is (A) 72 (B) 36 (C) 42 (D) 48 9. The sum of three numbers is 98. If the ratio between the first and second is 2 : 3 and that between the second and the third is 5 : 8, then the second number is (A) 30 (B) 20 (C) 58 (D) 48 10. The average height of 30 girls out of a class of 40 is 160 cm and that of the remaining girls is 156 cm. The average height of the whole class is (A) 158 cm (B) 158.5 cm (C) 159 cm (D) 157 cm 11. The average weight of 6 persons is increased by 2.5 kg when one of them whose weight is 50 kg is replaced by a new man. The weight of the new man is (A) i 65 kg (B) 75 kg (C) 76 kg (D) 60 kg From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 30 12. The average age of A, B C and D five years ago was 45 years. By including X, the present average age of all the five is 49 years. The present age of X is (A) 64 years (B) 48 years (C) 45 years (D) 40 years 13. The average salary of 20 workers in an office is Rs. 1900 per month. If the manager‘s salary is added, the average salary becomes Rs. 2000 per month. What is the manager‘s annual salary? (A) Rs. 24, 000 (B) Rs. 25,200 (C) Rs. 45,600 (D) None of these 14. The average weight of a class of 40 students is 40 kg. If the weight of the teacher be included, the average weight increases by 500 gm. The weight of the teacher is (A) 40.5 kg (B) 60 kg (C) 62 kg (D) 60.5 kg 15. In a Infosys test, a student scores 2 marks for every correct answer and loses 0.5 marks for every wrong answer. A student attempts all the 100 questions and scores 120 marks. The number of questions he answered correctly was (A) 50 (B) 45 (C) 60 (D) 68 16. The average of the first ten natural numbers is (A) 5 (B) 5.5 (C) 6.5 (D) 6 17. The average of the first ten whole numbers is (A) ****4.5 (B) 5 (C) 5.5 (D) 4 18. The average of the first ten even numbers is (A) 18 (B) 22 (C) 9 (D) 11 11 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 31 19. The average weight of a class of 30 students is 40 kg. If, however, the weight of the teacher is included, the average become 41 kg. The weight of the teacher is (A) 31 kg (B) 62 kg (C) 71 kg (D) 70 kg 20. 30 oranges and 75 apples were purchased for Rs. 510. If the price per apple was Rs. 2, then the average price of oranges was (A) Rs. 12 (B) Rs. 14 (C) Rs. 10 (D) Rs. 15 21. A batsman made an average of 40 runs in 4 innings, but in the fifth inning, he was out on zero. What is the average after fifth innings? (A) 32 (B) 22 (C) 38 (D) 49 22. The average weight of a school of 40 teachers is 80 kg. If, however, the weight of the principle be included, the average decreases by 1 kg. What is the weight of the principal? (A) 109 kg (B) 29 kg (C) 39 kg (D) None of these 23. The average age of Ram and Shyam is 20 years. Their average age 5 years hence will be (A) 25 years (B) 22 years (C) 21 years (D) 20 years 24. The average of 20 results is 30 and that of 30 more results is 20. For all the results taken together, the average is (A) 25 (B) 50 (C) 12 (D) 24 25. The average of 5 consecutive numbers is 18. The highest of these numbers will be (A) 24 (B) 18 (C) 20 (D) 22 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 32 26. Three years ago, the average age of a family of 5 members was 17 years. A baby having been born, the average of the family is the same today. What is the age of the baby? (A) 1 years (B) 2 years (C) 6 months (D) 9 months 27. Varun average daily expenditure is Rs. 10 during May, Rs. 14 during June and Rs. 15 during July. His approximate daily expenditure for the 3 months is (A) Rs. 13 approximately (B) Rs. 12 (C) Rs. 12 approximately (D) Rs. 10 28. A ship sails out to a mark at the rate of 15 km per hour and sails back at the rate of 20 km/h. What is its average rate of sailing? (A) 16.85 km (B) 17.14 km (C) 17.85 km (D) 18 km 29. The average temperature on Monday, Tuesday and Wednesday was 41 0 C and on Tuesday, Wednesday and Thursday it was 40 0 C. If on Thursday it was exactly 39 0 C, then on Monday, the temperature was (A) 42 0 C (B) 46 0 C (C) 23 0 C (D) 26 0 C 30. The average of 20 results is 30 out of which the first 10 results are having an average of 10. The average of the rest 10 results is (A) 50 (B) 40 (C) 20 (D) 25 31. A man had seven children. When their average age was 12 years a child aged 6 years died. The average age of the remaining 6 children is (A) 6 years (B) 13 years (C) 17 years (D) 15 years 32. The average weight of 35 students is 35 kg. If the teacher is also included, the average weight increases to 36 kg. The weight of the teacher is (A) 36 kg (B) 71 kg (C) 70 kg (D) 45 kg From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 33 33. The average of x, y and z is 45. x is as much more than the average as y is less than the average. Find the value of z. (A) 45 (B) 25 (C) 35 (D) 15 34. The average salary per head of all the workers in a company is Rs. 95. The average salary of 15 officers is Rs. 525 and the average salary per head of the rest is Rs. 85. Find the total number of workers in the workshop. (A) 660 (B) 580 (C) 650 (D) 46 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 34 PROFIT AND LOSS Business transactions have now-a-days become common feature of life. When a person deals in the purchase and sale of any item, he either gains or loses some amount generally. The aim of any business is to earn profit. The commonly used terms in dealing with questions involving sale and purchase are: Cost Price The cost price of an article is the price at which an article has been purchased. It is the abbreviated as C.P. Selling Price The selling price of an article is the price at which an article has been sold. It is abbreviated as S.P. Profit or Gain If the selling price of an article is more that the cost price, there is a gain or profit. Thus, Profit or Gain = S.P- C.P. Loss If the cost price of an article is greater than the selling price, the suffers a loss. Thus, Loss = C.P- S.P. Note that profit and loss are always calculated with respect to the cost price of the item. Illustration 1. (i)If C.P. = Rs. 235, S.P. = Rs. 240, then profit = ? (ii) If C.P. = Rs. 116, S.P. = Rs. 107, then loss = ? Solution (i) Profit = S.P.- C.P. =Rs. 240- 235 =Rs.5. (ii) Loss = C.P.- S.P. = Rs. 116- 107 =Rs.9. SOME BASIC FORMULAE 1. Gain on Rs. 100 is Gain per cent Gain% = (Gain × 100)/C.P Loss on Rs. 100 is Loss per cent Loss% = (Loss × 100)/C.P Illustration 2 The cost price of a shirt is Rs. 200 and selling price is Rs. 250. Calculate the % profit. Solution We have, C.P. = Rs. 200, S.P = Rs. 250. Profit = S.P.- C.P. = 250- 200 =Rs.50. Profit% = profit× 100/ C.P = 50× 100/ 250 = 25% Illustration 3 Anu bought a necklace for Rs. 750 and sold it for Rs. 675. Find her percentage loss. Solution Here, C.P. = 750, S.P. = Rs. 675. Loss= C.P- S.P. = 750-675 = Rs. 75. Loss% = Loss × 100/ C.P = 75× 100/ 750 = 10% 2. When the selling price and gain% are given: C.P = 100×S.P / (100+Gain%) 3. When the cost and gain per cent are given; S.P = (100+Gain%)×C.P/ 100 4. When the cost and loss per cent are given: S.P = (100-Loss%)×C.P / 100 5. When the selling price and loss per cent are given: C.P =(100)×S.P / (100-Loss%) Illustration 4 Mr. Sharma buys a cooler for Rs. 4500. For how much should he so that there is a gain of 8%? Solution We have, C.P. = Rs. 4500, gain% = 8% S.P = (100+Gain%/100)×C.P. = (100+ 8/ 100) × 4500 108/100 × 4500 = Rs. 4860 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 35 Illustration 5 By selling a fridge Rs. 7200, Pankaj loses 10%. Find the cost price of the fridge. Solution We have, S.P. = Rs. 7200, loss% = 10%. C.P =(100/100-Loss%)×S.P. = (100/100-10) × 7200 100/90 × 7200= Rs. 8000. Illustration 6 By selling a pen for Rs. 99, Mohan gains 12 ½ %. Find the cost price of the pen. Solution Here, S.P. = Rs. 99, gain% = 12 ½% or 25/2%. C.P =(100/100+Gain%)×S.P. = (100/100+25/2) ×99 = (100×2/ 225) ×99 =Rs. 88 SOME USEFUL SHORT-CUT METHODS 1. If a man buys x items for Rs. y and sells z items for Rs. w, then the gain or loss per cent made by him is (xw/zy -1) × 100%. Explanation S.P. of z items = Rs. w S.P. of x items = Rs. w/z x Net profit =w/z x-y. % profit = w/z x-y/y ×100% i.e.(xw/zy -1) ×100, which represent loss, if the result is negative. Note: In the case of gain per cent the result obtained bears positive sign whereas in the case of loss per cent the result obtained bears negative sign. How to remember: 1. Cross-multiply the numbers connected by the arrows (xw and zy) 2. Mark the directon of the arrows for crossmultiplicaton. The arrow going down forms the numerator while the arrow going up forms the denominator (xw/ zy). Illustration 7 If 11 oranges are bought for Rs. 10 and sold at 10 for Rs. 11 what is the gain loss%? Solution % profit= (xw/zy -1) × 100% = (11×11/10×10-1) ×100% = 21/100 ×100% = 21% Illustration 8 A fruit seller buys apples at the rate of Rs 12 per dozen and sells them at the rate of 15 for Rs.12. Find his percentage gain or loss. Solution % gain or loss = (xw/ zy -1) × 100% = (12× 12/15 ×12 -1) ×100% = -36/144 × 100% = -25% Since the sign is –ve, there is a loss of 25%. 2. If the cost price of m articles is equal to the selling price of n articles, then % gain or loss = ( m-n/n) ×100 [If m > n, it is gain and if m<n, it is loss] Explanation From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 36 Let the C.P. of one article be Re.1 C.P. of m articles = Rs. m×1=Rs. m S.P. of n articles = Rs. m S.P. of 1 articles = Rs. m/n Profit on 1 article = Rs.(m/n-1) i.e. Rs. (m-n/n) %profit = m-n/n × 100/1 i.e.(m-n/n) ×100. Illustration A shopkeeper professes to sell his goods on cost price but uses 800 gm, instead of 1kg. What is his gain %? Solution Here, cost price of 1000 is equal to selling price of 800 gm. % gain = (m-n/n) ×100 = (1000-800/800) × 100 = 200/800×100 =25% Illustration 10 If the selling price of 12 articles is equal to the cost price of 18 articles, what is the profit %? Solution Here, m = 10, n =12 Profit %= (m-n/n) ×100 (18-12/12) ×100 = 6/12 ×100= 50% EXERCISE 1. By selling a watch for Rs.495, a shopkeeper incurs a loss of 10%. Find the cost price of the watch for the shopkeeper. 1. 545/- 2. 550/- 3. 555/- 4. None 2. A cellular phone when sold for Rs.4600 fetches a profit of 15%. Find the cost price of the cellular phone. 1. 4300/- 2. 4150/- 3. 4000/- 4. 4500/- 3. A machine costs Rs.375. If it is sold at a loss of 20%, what will be its cost price as a percentage of its selling price? 1. 80% 2. 120% 3. 110% 4. 125% 4. A shopkeeper sold goods for Rs.2400 and made a profit of 25% in the process. Find his profit per cent if he had sold his goods for Rs.2040. 1. 6.25% 2. 7% 3. 6.20% 4. 6.5% 5. By selling bouquets for Rs.63, a florist gains 5%. At what price should he sell the bouquets to gain 10% on the cost price? 1. 66/- 2. 69/- 3. 72/- 4. 72.50/- 6. The cost price of a shirt and a pair of trousers is Rs.371. If the shirt costs 12% more than the trousers, find the cost price of the trouser. 1. 125/- 2. 150/- 3. 175/- 4. 200/- 7. ******A pet shop owner sells two puppies at the same price. On one he makes a profit of 20% and on the other he suffers a loss of 20%. Find his loss or gain per cent on the whole transaction. 1. Gain of 4% 2. No profit no loss 3. Loss of 10% 4. Loss of 4% 8. The marked price of a table is Rs.1200, which is 20% above the cost price. It is sold at a discount of 10% on the marked price. Find the profit per cent. 1. 10% 2. 8% 3. 7.5% 4. 6% 9. *****A shopkeeper marks the price of an articles at Rs.80. Find the cost price if after allowing a discount of 10% he still gains 20% on the cost price. 1. 53.33/- 2. 70/- 3. 75/- 4. 60/- From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 37 10. ****The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25% then the value of ―x‖ is. 1. 25 2. 18 3. 16 4. 15 11. A dozen pairs of gloves quoted at Rs.80 are available at a discount of 10%. Find how many pairs of gloves can be bought for Rs.24. 1. 4 2. 5 3. 6 4. 8 12. ******Find a single discount equivalent to the discount series of 20%, 10%, 5%. 1. 30% 2. 31.6% 3. 68.4% 4. 35% 13. *****How much percent more than the cost price should a shopkeeper mark his goods, so that after allowing a discount of 12.5% he should have a gain of 5% on his outlay? 1. 9.375 2. 16.66% 3. 20% 4. 25% 14. ****In order to maintain the price line, a trader allows a discount of 10% on the marked price of goods in his shop. However, he still makes a gross profit of 17% on the cost price. Find the profit percent he would have make on the selling price had he sold at he marked price. 1. 23.07 2. 30% 3. 21.21% 4. 25% 15. *****qq A whole-seller allows a discount of 20% on the list price to a retailer. The retailer sells at 5% discount on the list price. If the customer paid Rs.38 for an article, what profit is made by the retailer? 1. 10/- 2. 8/- 3. 6/- 4. None 16. *****The cost of production of a cordless phone set in 2002 is Rs.900, divided between material, labor and overheads in the ratio 3:4:2. If the cordless phone set is marked at a price that gives a 20% profit on the component of price accounted for by labor, what is the marked price of he set? 1. 980/- 2. 1080/- 3. 960/- 4. None 17. *****A shopkeeper sells an article at a profit of 10% and uses weights which are 20% less than the arrival weight the total gain earned by him will be 1. 30% 2. 88% 3. 37.5% 4. None of these 18. A man sells 5 articles forRs.15 and makes a profit of 20%. Find his gain or loss percent if he sells 8 such articles for Rs.18.40. 1.2.22% profit 2. 2.22% loss 3. 8% loss 4. 8% profit 19. The cost price of 50 mangoes is equal to the selling price of 40 mangoes. Find the percentage profit? 1. 20% 2. 25% 3. 30% 4. None 20. A makes an article for Rs.120 and sells it to B at a profit of 25%. B sells it to C who sells it for Rs.198, making a profit of 10%. What profit percent did B make? 1. 25% 2. 20% 3. 16.66% 4. 15% 21. *****A reduction of 10% in the price of sugar enables a housewife to buy 6.2kg more for Rs.279. Find the reduced price per kilogram. 1. 5/- 2. 4.5% 3. 4.05% 4. None 22. A man buys 50 kg of oil at Rs.10 per kilogram and another 40 kg of oil at Rs.12 kilogram and mixes them. He sells the mixture at the rate of Rs.11 per kilogram. What will be his gain percent if he is able to sell the whole lot? 1. 100/98% 2. 100(10/49)% 3. 10(1/49)% 4. None 23. *A shopkeeper sells sugar in such a way that the selling price of 950 gm is the same as the cost price of one kilogram. Find his gain percent. 1. 100/17% 2. 150/17% 3. 5(5/19)% 4. 1/19% 24. *A sold a table to B at a profit of 15%. Later on, B sold it back to A at a profit of 20%, thereby gaining Rs.69. How much did A pay for the table originally? 1. 300/- 2. 320/- 3. 345/- 4. 350/- From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 38 25. A color TV and a VCP were sold for Rs.12, 000 each. The TV was sold at a loss of 20% whereas the VCP was sold at a gain of 20%. Find gain or loss in the whole transaction. 1. 1200/- loss 2. 1000/- loss 3. 960/- loss 4. 1040/- loss 26. *A man sells a TV set for Rs.3450 and makes a profit of 15%. He sells another TV at a loss of 10%. If on the whole, he neither gains nor loses, find the selling price of the second TV set. 1. 4000/- 2. 4400/- 3. 4050/- 4. 4500/- 27. A man sells an article at 5% above its cost price. if he had bought it at 5% less than what he paid for it and sold it for Rs.2 less, he would have gained 10%. Find the cost price of the article. 1. 500/- 2. 360/- 3. 425/- 4. 400/- 28. A briefcase was sold at a profit of 10%. If its cost price was 5% less and it was sold for Rs.7 more, the gain would have been 20%. Find the cost price of the briefcase. 1. 175/- 2. 200/- 3. 225/- 4. 160/- 29. *A man buys two cycles for a total cost of Rs.900. By selling one for 4/5 of its cost and other for 5/4 of its cost, he makes a profit of Rs.90 on the whole transaction. Find the cost price of lower priced cycle? 1. 360/- 2. 250/- 3. 300/- 4. 420/- 30. A merchant bought two transistors, which together cost him Rs.480. He sold one of them at a loss of 15% and other at a gain of 19%. If the selling price of both the transistors are equal, find the cost of the lower priced transistor. 1. 300/- 2. 180/- 3. 200/- 4. 280/- 31. Two dealers X and Y selling the same model of refrigerator mark them under the same selling prices. X gives successive discounts of 25% and 5% and Y gives successive discounts of 16% and 12%. From whom is it more profitable to purchase the refrigerator? 1. From Y 2. From X 3. Indifferent between the two 4. cannot be determined 32. A firm dealing in furniture allows 4% discount on the marked price of each item. What price must be marked on a dining table that cost Rs.400 to assemble, so as to make a profit of 20%. 1. 475/- 2. 480/- 3. 500/- 4. 520/- 33. A shopkeeper allows a discount of 12.5% on the marked price of a certain article and makes a profit of 20%. If the article cost the shopkeeper Rs.210, what price must be marked on the article? 1. 280/- 2. 288/- 3. 300/- 4. None 34. `A watch dealer pays 10% custom duty on a watch that costs Rs.250 abroad. For how much should he mark it, if he desires to make a profit of 20% after giving a discount of 25% to the buyer? 1. 400/- 2. 440/- 3. 275/- 4. 330/- 35. *A dishonest dealer professes to sell at cost price but uses a 900 gram weight instead of 1 kg. Weight. Find the percent profit to the dealer. 1. 10% 2. 11.11% 3. 12.5% 4. None From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 39 SIMPLE AND COMPOUND INTEREST If P stands for Principal, R the rate per cent per annum, T the number of years, I the simple interest and A the amount, then 1. Simple Interest = Principal × Rate × Time 100 or, I = P × R × T 100 Principal = 100 × Simple Interest Rate × Time or, P = 100 × I R × T 3. Rate = 100 × Simple Interest Principal × Time or, R = 100 × I P × T 4. Time = 100 × Simple Interest Rate × Principal or, T = 100 × I R × P 5. Amount = Principal + Simple Interest = Principal + Principal × Rate × Time 100 Some Useful Short-Cut Methods 1. If a certain sum in T years at R% per annum amounts to Rs. A, then the sum will be P = 100 × A 100+R×T = Principal (1+(Rate ×Time)/100) or, A = P (1 + (R × T)/100) 2. The annual payment that will discharge a debt of Rs. A due in T years at R% per annum is Annual payment = Rs. (100A/(100T + RT(T – 1)/2)) 3. If a certain sum is invested in n types of investments in such a manner that equal amount is obtained on each investment where interest rates are R 1 , R 2 , R 3 , …, R n respectively and time periods are T 1 , T 2 , T 3 , …, T n respectively, then the ratio in which the amounts are invested is 1/100+R 1 T 1 : 1/100+R 2 T 2 1/100+R 3 T 3 : … 1/100+R n T n 4. If a certain sum of money becomes n times itself in T years at simple interest, then the rate of interest per annum is R = 100(n – 1)/T % 5. If a certain sum of money becomes n times itself at R% per annum simple interest in T years, then T = (n-1)/R) × 100 years. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 40 6. If a certain sum of money becomes n times itself in T years at a simple interest, then the time T in which it will become m times itself is given by T' = (m – 1/n – 1) × T years. 7. Effect of change of P, R and T on simple interest is given by the following formula: = Product of fixed parameter/100 × [difference of product of variable parameters] for example, if rate ® changes from R 1 to R 2 and P, T are fixed then Change in SI = PT/100 × (R 1 – R 2 ) Similarly, if principal (P) changes from P1 to P2 and R, T are fixed, then change in SI = RT/100 × (P 1 – P 2 ) Also, if rate ® changes from R1 to R2 and time (T) changes from T1 to T2 but principal (P) is fixed, then change in SI = P/100 × (R1T 1 – R 2 T 2 ). 8. If a debt of Rs. Z is paid in ‗n‘ number of instalments and if the value of each instalment is Rs. a, then the borrowed (debt) amount is given by Z = na + (Ra/100 × b) × n(n – 1)/2 Where, R = rate of interest per annum b = no. of instalments/year b = 1, when each instalment is paid yearly b = 2, when each instalment is paid half-yearly b = 4, when each instalment is paid quarterly b = 12, when each instalment is paid monthly. 9. If a certain sum of money P lent out at SI amounts to A 1 in T 1 years and to A 2 in T 2 years, then P = A 1 T 2 – A 2 T 1 T 2 – T 1 and, R = A 1 – A 2 /A 1 T 2 - A 2 T 1 × 100% 10. If a certain sum of money P lent out for a certain time T amounts to A 1 at R 1 % per annum and to A 2 at R 2 % per annum, then P = A 2 R 1 – A 1 R 2 /R 1 – R 2 and, T = A1 – A2 × 100 years. A2R1-A1R2 11. If an amount P 1 lent at simple interest rate of R 1 % per annum and another amount P 2 at simple interest rate of R 2 % per annum, then the rate of interest for the whole sum is R = (P 1 R 1 +P 2 R 2 /P 1 +P 2 ). 12. If a certain sum of money is lent out in n parts in such a manner that equal sum of money is obtained as simple interest on each part where interest rates are R 1 , R 2 , …, R n respectively and time periods are T 1 , T 2 , …, T n respectively, then the ratio in which the sum will be divided in n parts is given by 1/R 1 T 1 : 1/R 2 T 2 : …1/R n T n 13. When there is a change in principal (P), Rate (R) and Time (T), then the value of simple interest I also changes and is given by I 1 /I 2 = P 1 × R 1 × T 1 /P 2 × R 2 × T 2 ¬ A 1 – P 1 /A 2 – P 2 = P 1 × R 1 × T 1 /P 2 × R 2 × T 2 as I 1 = A 1 – P 1 and I 2 = A 2 – P 2 . From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 41 14. Out of a certain sum P, 1/a part is invested at R 1 %, 1/b part at R 2 % and the remainder (1-1/a-1/b) say 1/c part at R 3 %. If the annual income from all these investments is Rs. A, then the original sum is given by P = ((A × 100)/R 1 /A+R 2 /B+R 3 /C) EXERCISE 1. Interest obtained on a sum of Rs.5000 for 3 years is Rs.1500. Find the rate percent. 1. 8% 2. 9% 3. 10% 4. 11% 2. Rs.2100 is lent at compound interest of 5% per annum for 2 years. Find the amount after two years. 1. 2300/- 2. 2315.25/- 3. 2310/- 4. 2320/- 3. Find the difference between the simple and he compound interest at 5% per annum for 2 years on a principal of Rs.2000. 1. 5 2. 105 3. 4.5 4. None 4. After how many years will a sum of Rs.12,500 become Rs.17, 500 at the rate of 10% per annum? 1. 2 years 2. 3 years 3. 4 years 4. 5 years 5. What is the difference between the simple interest on a principal of Rs.500 being calculated at 5% per annum for 3 years and 4% per annum for 4 years? 1. 5/- 2. 10/- 3. 20/- 4. 40/- 6. What is the difference between compound interest and simple interest for the sum of Rs.2000 over a 2 year period if the compound interest is calculated at 20% and simple interest is calculated at 23% 1. 40/- 2. 46/- 3. 44/- 4. None 7. Find the compound interest on Rs.1000 at the rate of 20% per annum for 18 months when interest is compounded half-yearly? 1. 331/- 2. 1331/- 3. 320/- 4. None 8. Find the principal if compound interest is charged on the principal at the rate of 16 2/3% per annum for two years and the sum becomes Rs.196. 1. 140/- 2. 154/- 3. 150/- 4. None 9. The SBI lent Rs.1331 to the Tata group at a compound interest and got Rs.1728 after three years. What is the rate of interest charged if the interest is compounded annually? 1. 11% 2. 9.09% 3. 12% 4. 8.33% 10. Varun purchased a Maruti van for Rs.1, 96,000 and the rate of depreciation is 14 2/7% per annum. Find the value of he van after two years. 1. 1, 40, 000/- 2. 1, 44, 000/- 3. 1, 50, 000/- 4. None 11. Varun deposited Rs.8000 in ICICI Bank, which pays him 12% interest per annum compounded quarterly. What is the amount that he receives after 15 months? 1. 9274.2/- 2. 9228.8/- 3. 9314.3/- 4. 9338.8/- From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 42 12. What is he rate of simple interest for the first 4 years if the sum of Rs.360 becomes Rs.540 in 9 years and the rate of interest for the last 5 years is 6%? 1. 4% 2. 5% 3. 3% 4. 6% 13. Havishma makes a fixed deposit of Rs.20,000 with the Bank of India for a period of 3 years. If the rate of interest be 13% SI per annum charged half-yearly, what amount will he get after 42 months? 1. 27, 800/- 2. 28, 100/- 3. 29, 100/- 4. None 14. Varun makes a deposit of Rs.50,000 in the Punjab National Bank for a period of 2 ½ years. If the rate of interest is 12% per annum compounded half-yearly, find the maturity value of the money deposited by him. 1. 66, 911.27 2. 66, 123.34 3. 67, 925.95 4. 65, 550.8 15. Varun borrows Rs.1500 from two money lenders. He pays interest at the rate of 12% per annum for one loan and at he rate of 14% per annum for the other. The total interest he pays for the entire year is Rs.186.How much does he borrow at he rate of 12%? 1. 1200/- 2. 1300/- 3. 1400/- 4. 300/- 16. Two equal sums were borrowed at 8% simple interest per annum for 2 years and 3 years respectively. The difference in the interest was Rs.56. the sum borrowed were 1. 690/- 2. 700/- 3. 740/- 4. 780/- 17. In what time will Rs.500 give Rs.50 as interest at the rate of 5% per annum simple interest? 1. 2 years 2. 5 years 3. 3 years 4. 4 years 18. Shashikant derives an annual income of Rs.688.25 from Rs.10,000 invested partly at 8% p.a. and partly at 5% p.a. simple interest. How much of his money is invested at 5%? 1. 5, 000/- 2. 4225/- 3. 4, 8000/- 4. 3, 725/- 19. If the difference between the simple interest and compound interest on some principal amount at 20% per annum for 3 years is Rs.48, then the principal amount must be 1. 550/- 2. 500/- 3. 375/- 4. 400/- 20. Raju lent Rs.400 to Ajay for 2 years, and Rs.100 to Manoj for 4 years and received together from both Rs.60 as interest. Find the rate of interest, simple interest being calculated. 1. 5% 2. 6% 3. 8% 4. 9% 21. In what time will Rs.8000 amount to 40,000 at 4% per annum? (simple interest being reckoned) 1. 100 years 2. 50 years 3. 110 years 4. 160 years 22. What annual payment will discharge a debt of Rs.808 due in 2 years at 2% per annum? 1. 200/- 2. 300/- 3. 400/- 4. 350/- 23. A sum of money doubles itself in 5 years. In how many years will it become four fold (If interest is compounded)? 1.15 2. 10 3. 20 4. 12 24. Divide Rs.6000 into two parts so that simple interest on the first part for 2 years at 6% p.a. may be equal to the simple interest on the second part for 3 years at 8% p.a. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 43 1. 4000/-, 2000/- 2. 5000/-, 1000/- 3. 3000/-, 3000/- 4. None 25. A sum of money becomes 7/4 of itself in 6 years at a certain rate of simple interest. Find the rate of interest 1. 12% 2. 12(1/2)% 3. 8% 4. 14% 26. Samjay borrowed Rs.900 at 4% p.a. and Rs.1100 at 5% p.a. for the same duration. He had to pay Rs.364 in all as interest. What is the time period in years? 1. 5 years 2. 3years 3. 2 years 4. 4 years 27. If the difference between compound and simple interest on a certain sum of money for 3 years at 2% p.a. is Rs.604, what is the sum? 1. 500, 000 2. 450, 000 3. 510, 000 4. None 28. If a certain sum of money becomes double at simple interest in 12 years, what would be the rate of interest per annum? 1. 8(1/3) 2. 10 3. 12 4. 14 29. Three persons Amar, Akbar and Anthony invested different amounts in a fixed deposit scheme for one year at the rate of 12% per annum and earned a total interest of Rs.3, 240 at the end of the year. If the amount invested by Akbar is Rs.5000 more than the amount invested by Amar is Rs.5000 more than the amount invested by Amar and the amount invested by Anthony is Rs.2000 more than the amount invested by Akbar, what is the amount invested by Akbar? 1. 12, 000/- 2. 10, 000/- 3. 7, 000/- 4. 5000/- 30. A sum of Rs.600 amounts to Rs.720 in 4 years at Simple Interest. What will it amount to if the rate of interest is increased by 2% 1. 648/- 2. 768/- 3. 726/- 4. 792/- From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 44 TIME WORK In our daily life, we come across situations where we need to complete a particular job in a reasonable time. We have to complete the project earlier or later depending upon the needs. Accordingly, the men on duty have to be increased or decreased, i.e. the time allowed and the men engaged for a project are inversely proportional to each other, i.e. the more the number of men involved, the lesser is the time required to finish a job. We also come across situations where time and work or men and work are in direct proportion to each other. For solving problems on time and work, we follow the following general rules: 1. If ‗A‘ can do a piece of working ‗A‘ will finish 1/nth work in one day. 2. If 1/n of a work is done by ‗A‘ in one day, then ‗A‘ will take n days to complete the full work. 3. If ‗ A‘ does 1/nth of a work in one hour then to complete the full work, ‗A‘ will take n/m hours. 4. If ‗ A‘ does three times faster work than ‗B‘ then ratio of work done by A and B is 3:1 and ratio of time taken by A and B is1: 3. 5. A, B and C can do a piece of work in T 1 , T 2 and T 3 , days, respectively. If they have worked for D 1 , D 2 and D 3 days respectively, then Amount of work dine by A= D 1 /T 1 Amount of work dine by B= D 2 /T 2 And, Mount of work done by C=D 3 /T 3 Also, the amount of work done by A, B and C together = D 1 /T 1 +D 2 /T 2 +D 3 /T 3 Which will be equal to 1, if the work is complete? SOME USEFUL SHORT-CUT METHODS 1. If A can do a piece do a piece of work in X days and B can do the same work in Y days, then both of them working together will do the same work in XY/ X+Y days. Explanation A‘s 1 day‘s work = 1/X B‘s 1 day‘s work = 1/Y Then,(A+B)‘s 1 day‘s work = 1/X+1/Y= X+Y/XY A and B together can complete the work on = XY/ X+Y days. Illustration 1 A can finish a piece of work by working alone in 6 days and B, while works alone, can finish the same work in 12days. If both of them work together, then in how many days, the work will be finished? Solution Here, X= 6 and y = 12. Working together, A and B will complete the work in = XY/ X+Y days = 6×12/6+12 days, i.e. 4days. 2. If A, B and C, while working alone, can complete a work in X, Y and Z days respectively, then they will together complete the work in XYZ/XY+YZ+ZX days. Explanation A‘s 1 day‘s work = 1/X B‘s 1 day‘s work = 1/Y C‘s 1 day‘s work = 1/Z (A+B+C)‘s 1 days work =1/X+1/Y+1/Z = (XY+YZ+ZX)/XYZ. So, A, B and C together can complete the work in = (XYZ/ XY+YZ+ZX) days. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 45 Illustration 2 A,B and C can complete a piece of work in 10, 15 and 18 days, respectively. In how many days would all of them complete the same work working together? Solution Here, X=10, Y=15 and Z= 18. Therefore, the work will be completed in = XYZ/ XY+YZ+ZX days. = 10×15×18/10 ×15 +15 ×18+18 ×10 days, i.e. 2700/600 or, 4 ½ days. 3. Two persons A and B, working together , can complete a piece of work in X days. If A, working alone, can complete the work in XY/Y-X days. Explanation A and B together can complete the work in X days. (A+B)‘s 1 day‘s work = 1/X Similarly, A‘s 1day‘s work= 1/Y There fore, B‘s 1 day‘s work =1/X-1/Y= Y-X/XY B alone can complete the work in (XY/Y-X)days B alone will complete the work in Illustration 3 A and B working together take 15 days to complete a piece of work. If A alone can do this work in 20 days, how long would B take to complete the same work? Solution Here, X = 15, and Y=20. = XY/Y-X days =15×20/20-15, i.e. 60 days 4. If A and B, working together, can finish a piece of work in X days, B and C in Y days, then a) A, B and C working together, will finish the job in (2XYZ/XY+YZ- ZX) days. b) A alone will finish the job in (2XYZ/XY+YZ- ZX) days. c) B alone will finish the job in (2XYZ/ZX+XY- YZ) days. Explanation (A+B)‘s 1 day‘s work = 1/X (B+C)‘s 1 day‘s work = 1/Y (C+A)‘s 1 day‘s work = 1/Z So, [(A+B) + (B+C)+ (C+A)]‘s 1 day‘s work = 1/X+1/Y+1/Z or, 2(A+B+C)‘s 1day‘s work = (1/X+1/Y+1/Z) or, (A+B+C)‘s 1day‘s work = ½ (1/X+1/Y+1/Z) i.e (XY+YZ-XZ/2XYZ) A, B and C working together, will complete the work in (2XYZ/XY+ZX-XY) days. Also, A‘s 1 days work – (A+B+C)‘s 1day‘s work –(B+C)‘s 1 days work = ½ (1/X+1/Y+1/Z)- 1/Y = ½ (1/X-1/Y+1/Z) = XY+YZ-XZ/2XYZ So. A alone can do the work in (2XYZ/XY+YZ+XZ) days Similarly, B alone can do the work in (2XYZ/YZ+XY+XY) days and C alone can do the work in(2XYZ/ZX+XY+YZ) days. Illustration 4 A and B can do a piece of work in 12 days, B and C 15 days, C and A in 20 days. How long would each take separately to do the same work? Solution Here, X = 12, Y=15 and Z=20. A alone can do the work in = 2XYZ/XY+YZ-ZX = 2×12×15×20/12×15+15×20-20×12 days. or, 7200/240, i.e. 30 days. B alone can do the work in = 2XYZ/ZY+ZX-XY days = 2×12×15×20/15×20+20×12-12×15 days or, 7200/360, i.e. 20 days. C alone can do the work in = 2XYZ/ZX+XY-YZ = 2×12×15×20/20×12+12×15-15×20 or, 7200/120, i.e. 60 days. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 46 5. (a) If A can finish a work in X days and B is k times efficient than A, then the time taken by both A and B working together to complete the work is x/1+k. (b) If A and B working together can finish a work in X days and B is k times efficient than A, then the time taken by (i) A, working alone, to complete the work is (k+1)X. (ii) B, working alone, to complete the work is (k+1/k)X. Illustration 5 Harbans Lal can do a piece of work in 24 days. If Bansi Lal works twice as fast as Harbans Lal, how long would they take to finsh the work working together? Solution Here, X =24 and k=2. Time taken by Harbans Lal and Bansi Lal, woking together, to complete the work = (X/1+k)days. = (24/1+2) days, i.e. 8 days Illustration 6 A and B together can do a piece of work in 3 days. If A does thrice as much work as B in a given time, find how long A lone would take to do the work? Solution Here, X = 3 and k = 3. Time taken by A, working alone, to complete the work = (k+1/k) X = (3+1/3)3 = 4days 6. If A working alone takes a days more than A and B working alone takes b days more than A and B together, then the number of days taken by A and B, working together, to finish a job is given by \ab. Illustration 7 A alone would take 8 hours more to complete the job than if both A and B worked together. If B worked alone, he took 4 ½ hours more to complete the job than A and B worked together. What time would they take if both A and B worked together? Solution Here, a = 8 and b = 9/2. Time taken by A and B, working together, to complete the job = \ ab days = \ 8×9/2 or, 6days. EXERCISE 1. Nishu and Archana can do a piece of work in 10 days and Nishu alone can do it in 12 days. In how many days can Archana do it alone? 1. 60 days 2. 30 days 3. 50 days 4. 45 days 2. Baba alone can do a piece of work in 10 days. Anshu alone can do it in 15 days. If the total wages for the work is Rs.50. How much should Baba be paid if they work together for the entire duration of his work? 1. 30/- 2. 20/- 3. 50/- 4. None 3. 4 men and 3 women finish a job in6 days, and 5 men and 7 women can do the same job in 4 days. How long will 1 man and 1 woman take to do the work? 1. 22(2/7) days 2. 25(1/2) days 3. 5(1/7) days 4. 12(7/22) days 4. A can do a piece of work in 10 days and B can do the same work in 20 days. With the help of C, they finish the work in 5 days. How long will it take for C alone to finish the work? 1. 20 days 2. 10 days 3. 35 days 4. 15 days 5. A can do a piece of work in 20 days. He works at it for 5 days and then B finishes it in 10 more days. In how many days will A and B together finish the work? 1. 8 days 2. 10 days 3. 12 days 4. 6 days 6. A and B undertake to do a piece of work for Rs.100. A can do it in 5 days and B can do it in 10 days. With the help of C, they finish it in 2 days. How much should C be paid for his contribution? 1. 40/- 2. 20/- 3. 60/- 4. 30/- From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 47 7. *Twenty workers can finish a piece of work in 30 days. After how many days should 5 workers leave the job so that the work is completed in 35 days? 1. 5 days 2. 10 days 3. 15 days 4. 20 days 8. Arun and Vinay together can do a piece of work in 7 days. If Arun does twice as much work as Vinay in a given time, how long will Aruna alone take to do the work? 1. 6.33 days 2. 10.5 days 3. 11 days 4. 72 days 9. X number of men can finish a piece of work in 30 days. If there were 6 men more, the work could be finished in 10 days less. What is the original number of men? 1. 10 2. 11 3. 12 4. 15 10. Sashi can do a piece of work in 25 days and Rishi can do it in 20 days. They work for 5 days and then Sashi goes away. In how many more days will Rishi finish the work? (11 days) 1. 10 days 2. 12 days 3. 14 days 4. None 11. Raju can do a piece of work in 10 days, Vicky in 12 days and Tinku in 15 days. They all start the work together, but Raju leaves after 2 days and Vicky leaves 3 days before the work is completed. In how many days is the work completed? 1. 5 days 2. 6 days 3. 7 days 4. 8 days 12. Manoj takes twice as much time as Anjay and thrice as much as Vijjay to finish a piece overwork. Together they finish the work in 1 day. What is the time takes by Manoj to finish the work? 1. 6 days 2. 3 days 3. 2 days 4. None 13. In a company XYZ Ltd. A certain number of engineers can develop a design in 40 days. If there were 5 more engineers, it could be finished in 10 days less. How many engineers were there in the beginning? 1. 18 2. 20 3. 25 4. 15 14. If 12 men and 16 boys can do a piece of work in 5 days and 13 men and24 boys can do it in 4 days, compare the daily work done by a man with that done by a boy? 1. 1:2 2. 1:3 3. 2:1 4. 3:1 15. A can do a work in 10 days and B can do the same work in 20 days. They work together for 5 days and then a goes away. In how many more days will B finish the work? 1. 5 days 2. 6.5 days 3. 10 days 4. 8 1/3 days 16. 30 men working 5h a day can do a work in 16 days. In how many days will 20 men working 6h a day do the same work? 1. 22 1/2 days 2. 20 days 3. 21 days 4. None 17. ajay and Vijay undertake to do a piece of work for Rs. 200. ajay alone can do it in 24 days while Vijay alone can do it in 30 days. With the help of pradeep they finish the work in 12 days. How much should pradeep get for his work 1. 20/- 2. 100/- 3. 180/- 4. 50/- 18. *15 men could finish a piece of work in 210 days. But at the end of 100 days, 15 additional men are employed. In how many more days will the work be completed? 1. 80 days 2. 60 days 3. 55 days 4. 50 days 19. Ajay, vijay and Sanjay are employed to do a piece of work for Rs. 529. Ajay and Vijay together are supposed to do 19/23 of the work. Vijay & Sanjay together 8/23 of the work how much should Ajay be paid? 1. 245/- 2. 295/- 3. 300/- 4. 345/- 20. *In a fort there was sufficient food for 200 soldiers for 31 days. After 27 days 120 soldiers left the fort. For how many extra days will the rest of the food last for the remaining soldiers? 1. 12 days 2. 10 days 3. 8 days 4. 6 days 21. Ajay and Vijay cam do a piece of work in 28 days. With the help of Manoj, they can finish it in 21days. How long will Manoj take to finish the work alone. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 48 1. 84 days 2. 80 days 3. 75 days 4. 70 days 22. Ajay and Vijay together can do a piece of work in 6 days. Ajay alone does it in 10 days. What time does Vijay require to do it alone? 1. 20 days 2. 15 days 3. 25 days 4. 30 days 23. A cistern is normally filled in 5 hours. However, it takes 6 hours when there is leak in its bottom. If the cistern is full, in what time shall the leak empty it? 1. 6h 2. 5h 3. 30h 4. 15h 24. *Pipe A and B running together can fill a cistern in 6 minutes. If B takes 5 minutes more than A to fill the cistern, then the time in which A and B will fill the cistern separately will be respectively? 1. 15 min, 20 min 2. 15 min, 10 min 3. 10 min, 15 min 4. 25 min, 20 min 25. A can do a work in 18 days, B in 9 days and C in 6 days. A and B start working together and after 2 days C joins them. In how many days will the job be completed? 1. 4.33 days 2. 4 days 3. 4.66 days 4. 5 days 26. 24 men working 8 h a day can finish a work in 10 days. Working at a rate of 10 h a day, the number of men required to finish the work in 6 days is 1. 30 2. 32 3. 34 4. 36 27. A certain job was assigned to a group of men to do it in 20 days. But 12 men did not turn up for the job and the remaining men did the job in 32 days. The original number of men in group was 1. 32 2. 34 3. 36 4. 40 28. *12 men complete a work in 18 days. 6 days after they had started working, 4 men join them. How many more days will all of them to complete the remaining work? 1. 10 days 2. 12 days 3. 15 days 4. 9 days 29. A cistern is normally filled in 6 h but takes 4H longer to fill because of a leak in its Bottom. If the cistern is full, the leak will empty it in how much time? 1. 15 h 2. 16 h 3. 20 h 4. None 30. There are two pipes in a tank. Pipe A is for filling the tank and Pipe B is for emptying the tank. If A can fill the tank in 10 hours and B can empty the tank in 15 hours hen find how many hours will nit take to completely fill a half empty tank? 1. 30 hours 2. 15 hours 3. 20 hours 4. 33.33 hours 31. Abbot can do some work in 10 days, Bill can do it in 20 days and Clinton can do it in 40 days. They start working in turn with Abbot starting to work on the first day followed by Bill on the second day and by Clinton on the third day and again by Abbot on the fourth day and so on till the work is completed fully. Find the time taken to complete the work fully? 1. 16 days 2. 15 days 3. 17 days 4. 16.5 days 32. A, B and C can do some work in 36 days. A and B together do twice as much work as C alone and A and C together can do thrice as much work as B alone. Find the time taken by C to do the whole work? 1. 72 days 2. 96 days 3. 108 days 4. 120 days 33. *There are three Taps A, B and C in a tank. They can fill the tank in 10 hrs, 20 hrs and 25 hrs respectively. At first, all of them are opened simultaneously. Then after 2 hours, tap C is closed and A and B are kept running after the 4 th hour, tap B is also closed. The remaining work is done by Tap A alone. Find the percentage of the work done by Tap A by itself. 1. 32% 2. 52% 3. 75% 4. None 34. Two taps are running continuously to fill a tank. The 1 st tap could have filled it in 5 hours by itself and the second one by itself could have filled it in 20 hours. But the operator failed to realize that there was a leak in the tank from the beginning which caused a delay of one hour in the filling of the tank. Find the time in which the leak would empty a filled tank? 1. 15 hours 2. 20 hours 3. 25 hours 4. 40 hours 24 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 49 35. A can do some work in 24 days, B can do it in 32 days and C can do it in 60 days. They start working together. A left after 6 days and B left after working for 8 days. How many more days are required to complete the whole work? 1. 30 2. 25 3. 22 4. 20 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 50 Time and Distance The terms ‗Time‘ and ‗Distance‘ are related to the speed of a moving object. Speed: We define the speed of an object as the distance covered by it in a unit time interval. It is obtained by dividing the distance covered by the object, by the time it takes to cover that distance. Thus, Speed = Distance traveled/ Time taken Notes: 1. If the time taken is constant, the distance traveled is proportional to the speed, that is, more the speed; more the distance traveled in the same time. 2. If the speed is constant, the distance traveled is proportional to the time taken, that is, more the distance traveled; more the time taken at the same speed. 3. If the distance traveled is constant, the speed is inversely proportional to the time taken, that is, more the speed; less the time taken for the same distance traveled. SOME BASIC FORMULAE 1. Speed = Distance/Time 2. Distance – Speed × Time 3. Time = Distance/speed Units of Measurement Generally if the distance is measured in kilometre, we measure time in hours and speed in kilometre per hour and is written as km/hr and if the distance is measured in metre then time is taken in second and speed in metre per second and is written as m/sec. Conversion of Units One kilometre/hour = 1000metre/60×60 seconds = 5/18 m/sec. One metre/second = 18/5km/hr. Thus, x km/hr =(x ×5/18) m/sec. and, x m/sec. =(x ×18/5) km/hr. Illustration 1 Calculate the speed of a train which covers a distance of 150 km in 3 hours. Solution Speed =Distance covered/Time taken= 150/3 = 50km/hr Illustration 2 How long does a train 100 metres long running at the rate of 40 km/hr take to cross a telegraphic pole? Solution In crossing the pole. the train must travel its own length. Distance traveled is 100 metres. Speed = 40 km/hr. = 40×1000/ 60×60 = 100/9m/sec. Time taken to cross the pole = 100/(100/9) = 9seconds. Illustration 3 A train running at a speed of 90 km/hr passes a pole on the platform in 20 seconds. Find the length of the train in metres. Solution Speed of the train = 90km/hr =90×5/18 = 15 m/sec. Length of the train = Speed of the train × time taken in crossing the pole =25× 20=500m. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 51 SOME USEFUL SHORT-CUT METHODS 1. (a) If A covers a distance d 1 km at s 1 km/hr and then d 2 km at s 2 km/hr, then the average speed during the whole journey is given by *(1) Average speed = s 1 s 2 (d 1 + d 2 )/s 1 d 2 + s 2 d 1 km/hr. (b) if A goes from X to Y at s 1 km/hr. and comes back from Y to X at s 2 km/hr., then the average speed during the whole journey is given by Average speed 2s 1 s 2 /s 1 +s 2 Explanation a) Time taken to travel d 1 km at s 1 km/hr is t 1 = d 1 /s 1 hr. Time taken to travel d 2 km at s 2 km/hr is t 2 = d 2 /s 2 hr. Total time taken = t 1 + t 2 = (d 1 /s 1 +d 2 /s 2 )hr = (s 1 d 2 + s 2 d 2 /s 1 d 1 /s 1 s 2 )hr Total distance covered = (d 1 +d 2 )km. Therefore, Average speed =Total distance covered / Total time taken = s 1 s 2 (d 1 +d 2 )/(s 1 d 2 +s 2 d 1 )km/hr …(i) b) Let the distance from X to Y be d km. Take d 1 = d 2 =d in (i), we get Average speed = 2ds 1 s 2 /d(s 1 +s 2 ) = 2s 1 s 2 /s 1 s 2 . *(2) Illustration 4 A ship sails to a certain city at the speed of 15 knots/hr and sails back to the same point at the rate of 30 knots/hr. What is the average speed for the whole journey? Solution Here, s 1 = 15 and s 2 = 30. Average speed = 2s 1 s 2 /s 1 +s 2 = 2×15×30/ 15+30 =20knots/hr. 2. A person goes certain distance(A to B) at a speed of s 1 km/hr. and returns back (B to A) at a speed of s 2 km/hr. If the takes T hours in all, the distance between A and B is T(s 1 s 2 /s 1 +s 2 ) Explanation Let the distance between A and B be d km Time taken during onward journey = t 1 = d/s 1 hrs Time taken during return journey = t 2 = d/s 2 hrs Total time taken during the entire journey is T = t 1 +t 2 = d/s 1 +d/s 2 = d(s 1 +s 2 )/s 1 s 2 d = T(s 1 s 2 /s 1 +s 2 ) Thus, the distance between A and B is = T(s 1 s 2 /s 1 +s 2 ) = Total time taken × Product of two speeds/Sum of two speeds *(3) Illustration 5 A boy goes to school with the speed of 3 km an hour and returns with a speed of 2 km/hr. if he takes 5 hours in all, find the distance in km between the village and the school. Solution Here, s 1 =3, s 2 = 2 and T = 5. The distance between the village and the school= T (s 1 s 2 /s 1 +s 2 ) =5(3×2/3+2) = 6km. 3. If two persons A and B start at the same time from two points P and Q towards each other and after crossing they take T 1 and T 2 hours in reaching Q and P reactively, then *(4) A’s speed/B’s speed =\T 2 / \T 1 . Explanation Let the total distance between P and Q be d km. Let the speed of A be s 1 km/hr and that of B be s 2 km/hr. Since they are moving in opposite directions, their relative speed is (s 1 +s 2 )km/hr. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 52 EXERCISE 1. *The Sinhagad Express left Pune at noon sharp. Two hours later, the Deccan Queen started from Pune in the same direction. The Deccan Queen overtook the Sinhagad Express at 8 p.m. Find the average speed of the two trains over this journey if the sum of their average speeds is 70 km/h. (A) 34.28 km/h (B) 35 km/h (C) 50 km/h (D) 12 km/h 2. Walking at ¾ of his normal speed, Abhishek is 16 minutes late in reaching his office. The usual time taken by him to cover the distance between his home and his office is (A) 48 minutes (B) 60 minutes (C) 42 minutes (D) 62 minutes 3. *Two trains for Bombay leave Delhi at 6 a.m. and 6 : 45 am and travel at 100 kmph and 136 kmph respectively. How many kilometers from Delhi will the two trains be together (A) 262.4 km (B) 260 km (C) 283.33 km (D) None of these 4. Two trains. Calcutta Mail and Bombay Mail, start at the same time from stations Calcutta and Bombay respectively towards each other. After passing each other, they take 12 hours and 3 hours to reach Bombay and Calcutta respectively. If the Calcutta Mail is moving at the speed of 48 km/h, the speed of the Bombay Mails is (A) 24 km/h (B) 22 km/h (C) 21 km/h (D) 96 km/h 5. Without stoppage, a train travels a certain distance with an average speed of 60 km/h, and with stoppage, it covers the same distance with an average speed of 40 km/h. On an average, how many minutes per hour does the train stop during the journey? (A) 20 min/h (B) 15 min/h (C) 10 min/h (D) 5 min/h 6. *Rishikant, during his journey, travels for 20 minutes at a speed of 30 km/h, another 30 minutes at a speed of 50 km/h, and 1 hour at a speed of 50 km/h and 1 hour at a speed of 60 km/h. What is the average velocity? (A) 51.18 km/h (B) 63 km/h (C) 39 km/h (D) 48 km/h 7. Narayan Murthy walking at a speed of 20 km/h reaches his college 10 minutes late. Next time he increases his speed by 5 km/h, but finds that he is still late by 4 minutes. What is the distance of his college from his house. (A) 20 km (B) 6 km (C) 12 km (D) None of these From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 53 8. Manish travels a certain distance by car at the rate of 12 km/h and walks back at the rate of 3 km/h. The whole journey took 5 hours. What is the distance he covered on the car? (A) 12 km (B) 30 km (C) 15 km (D) 6 km 9. A railway passenger counts the telegraph poles on the rail road as he passes them. The telegraph poles are at a distance of 50 metres. What will be his count in 4 hours, if the speed of the train is 45 km per hour. (A) 600 (B) 2500 (C) 3600 (D) 5000 10. Two trains A and B start simultaneously in the opposite direction from two points A and B and arrive at their destinations 9 and 4 hours respectively after their meeting each other. At what rate does the second train B travel if the first train travels at 80 km per hour. (A) 60 km/h (B) 100 km/h (C) 120 km/h (D) None of these 11. ***Vinay fires two bullets from the same place at an interval of 12 minutes but Raju sitting in a train approaching the place hears the second report 11 minutes 30 seconds after the first. What is the approximate speed of train (if sound travels at the speed of 330 metre per second)? (A) 660/23 m/s (B) 220/7 m/s (C) 330/23 m/s (D) 110/23 m 12. A car driver, driving in a fog, passes a pedestrian who was walking at the rate of 2 km/h in the same directions. The pedestrian could see the car for 6 minutes and it was visible to him up to a distance of 0.6 km. What was the speed of the car? (A) 30 km/h (B) 15 km/h (C) 20 km/h (D) 8 km/h 13. *Harsh and Vijay move towards Hosur starting from IIM, Bangalore, at a speed of 40 km/h and 60 km/h respectively. If Vijay reaches Hosur 200 minutes earlier then Harsh, what is the distance between IIM. Bangalore, and Hosur? (A) 600 km (B) 400 km (C) 900 km (D) 200 km 14. A cyclist moving on a circular track of radius 100 metres completes one revolution in 2 minutes. What is the average speed of cyclist (approximately)? (A) 314 m/minute (B) 200 m/minute (C) 300 m/minute (D) 900 m/minute From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 54 15. A person traveled a distanced of 200 kilometre between two cities by a car covering the first quarter of the journey at a constant speed of 40 km/h and the remaining three quarters at a constant speed of x km/h. If the average speed of the person for the entire journey was 53.33 km/h what is the value of x. (A) 55 km/h (B) 60 km/h (C) 70 km/h (D) 80 km/h 16. A car travels 1/3 of the distance on a straight road with a velocity of 10 km/h, the next 1/3 with a velocity of 20 km/h and the last 1/3 with a velocity of 60 km/h. what is the average velocity of the car for the whole journey? (A) 18 km/h (B) 10 km/h (C) 20 km/h (D) 15 km/h 17. Two cars started simultaneously toward each other from town A and B, that are 480 km apart. It took the first car traveling from A to B 8 hours to cover the distance and the second car traveling from B to A 12 hours. Determine at what distance from A the two cars meet. (A) 288 km (B) 200 km (C) 300 km (D) 196 km 18. Walking at ¾ of his usual speed, a man is 16 minutes late for his office. The usual time taken by him to cover that distance is (A) 48 minutes (B) 60 minutes (C) 42 minutes (D) 62 minutes 19. *Two trains for Patna leave Delhi at 6 a.m. and 6:45 a.m. and travel at 98 kmph and 136 kmph respectively. How many kilometres from Delhi will the two trains meet? (A) 262.4 km (B) 260 km (C) 200 km (D) None of these 20. Two trains A and B start from station X to Y, Y to X respectively. After passing each other, they take 12 hours and 3 hours to reach Y and x respectively. If train A is moving at the speed of 48 km/h, the speed of train B is (A) 24 km/h (B) 22 km/h (C) 21 km/h (D) 20 km/h 21. X and Y are two stations 600 km apart. A train starts from X and moves towards Y at the rate of 25 km/h. Another train starts from Y at the rate of 35 km/h. How far from X they will cross each other (A) 250 km (B) 300 km (C) 450 km (D) 475 km From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 55 22. ***A motorboat went downstream for 28 km and immediately returned. It took the boat twice as long to make the return trip. If the speed of the river flow were twice as high, the trip downstream and back would take 672 minutes. Find the speed of the boat in still water and the speed of the river flow. (A) 9 km/h, 3 km/h (B) 9 km/h, 6 km/h (C) 8 km/h, 2 km/h (D) 12 km/h, 3 km/h 23. A train requires 7 seconds to pass a pole while it requires 25 seconds to cross a stationary train which is 378 metres long. Find the speed of the train. (A) 75.6 km/h (B) 75.4 km/h (C) 76.2 km/h (D) 21 km/h 24. A boat sails downstream from point A to point B, which is 10 km away from A, and then returns to A. If he actual speed of the boat (in still water) is 3 km/h, the trip from A to B takes 8 hours less than that from B to A. What must the actual speed of the boat for the trip from A to B to take exactly 100 minutes? (A) 1 km/h (B) 2 km/h (C) 3 km/h (D) 4 km/h 25. A boat goes 40 km upstream in 8 h and a distance of 49 km downstream in 7 h. The speed of the boat in still water is (A) 5 km/h (B) 5.5 km/h (C) 6 km/h (D) 6.5 km/h 26. * Two trains are running on parallel line in the same direction at speeds of 40 kmph and 20 kmph respectively. The faster train crosses a man in the second train in 36 seconds. The length of the faster train is (A) 200 metres (B) 185 metres (C) 225 metres (D) 210 metres 27. *Two trains pass each other on parallel lines. Each train is 100 metres long. When they are going in the same direction, the faster one takes 60 seconds to pass the other completely. If they are going in opposite directions they pass each other completely in 10 seconds. Find the speed of the slower train in km/h. (A) 30 km/h (B) 42 km/h (C) 48 km/h (D) 60 km/h 28. Two trains are traveling in the same direction at 50 km/h and 30 km/h respectively. The faster train crosses a man in the slower train in 18 seconds. Find the length of the faster train. (A) 0.1 km (B) 1 km (C) 1.5 km (D) 1.4 km From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 56 29. Two trains for Howrah leave Muzaffarpur at 8:30 a.m. and 9:00 a.m. respectively and travel at 60 km/h and 70 km/h respectively. How many kilometres from Muzaffarpur will the two trains meet? (A) 210 km (B) 180 km (C) 150 km (D) 120 km 30. Without stoppage, a train travels at an average speed of 75 km/h and with stoppages it covers the same distance at an average speed of 60 km/h. How many minutes per hour does the train stop? (A) 10 minutes (B) 12 minutes (C) 14 minutes (D) 18 minutes 31. Vijay can row a certain distance downstream in 6 h and return the same distance in 9 h. If the stream flows at the rate of 3 km/h, find the speed of Vijay in still water. (A) 12 km/h (B) 13 km/h (C) 14 km/h (D) 15 km/h 32. ****In a stream that is running at 2 km/h, a man goes 10 km upstream and comes back to the starting point in 55 minutes. Find the speed of the man in still water. (A) 20 km/h (B) 22 km/h (C) 24 km/h (D) 28 km/h 33. ****A motorboat went down the river for 14 km and then up the river for 9 km. It took a total of 5 hours for the entire journey. Find the speed of the river flow if the speed of the boat in still water is 5 km/h. (A) 1 kmph (B) 1.5 kmph (C) 2 kmph (D) 3 kmph 34. A motorboat whose speed in still water is 10 km/h went 91 km downstream and then returned to its starting point. Calculate the speed of the river flow if the round trip took a total of 20 hours. (A) 3 km/h (B) 4 km/h (C) 6 km/h (D) 8 km/h 35. A motorboat whose speed in still water is 15 kmph goes 30 km downstream and comes back in a total 4 hours 30 min. Determine the speed of the stream. (A) 2 kmph (B) 3 kmph (C) 4 kmph (D) 5 kmph From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 57 AGE PROBLEMS Problems based on ages are generally asked in most of the company examinations, To solve these problems, the knowledge of linear equations is essential. In such problems, there may be three situations: i. Age some years ago ii. Present age iii. Age some years hence Two of these situations are given and it is required to find the third. The relation between the age of two persons may also be given. Simple linear equations are framed and their solutions are obtained. Sometimes, shortcut methods given below are also helpful in solving such problems. SOME USEFUL SHORT-CUT METHODS 1. If the age of A, t years ago, was n times the age of B and at present A‘s age is n 2 times that of B, then A‘s present age = (n 1 -1/n 1 -n 2 ) n 2 t years and, B‘s present age =(n 1 -1/n 1 -n 2 )t years Explanation Let the present age of B be x years. Then, the present age of A =n 2 x years Given, t years ago, n 1 (x-t)=n 2 x-t or, (n 1 -n 2 )x = (n 1 -1) t or, x=(n 1 -1/n 1 -n 2 ) t years. Therefore, B‘s present age = (n 1 -1/n 1 -n 2 ) t years. And, A‘s present age =(n 1 -1/n 1 -n 2 )n 2 t years Illustration 1 The age of father is 4 times the age of his son. If 5 years age father‘s age was 7 times the age of his son at that time, what is father‘s present age? Solution The father‘s present age = (n 1 -1/n 1 -n 2 )n 2 t [Here, n 1 =7, n 2 =4 and t=5] =(7-1/7-4) 4×5 = 6×4×5/3 = 40years. 2. The present age of A is n 1 times the present age of B. If t years hence, the age of A would be n 2 time that of B, then A‘s present age = (n 1 -1/n 1 -n 2 )n 2 t years and B‘s present age = (n 1 -1/n 1 -n 2 )t years Explanation Let the present age of B be x years. Then, the present age of A= n 1 x Given, t years hence, (n 1 x +t)=n 2 (x+t) or (n 1 -n 2 )x = (n 2 -1)t or, x=(n 2 -1/n 1 -n 2 )t Therefore, B‘s present age =(n 2 -1/n 1 -n 2 )n 1 t years and, A‘s present age =(n 2 -1/n 1 -n 2 )n 1 t years Illustration 2 The age of Mr. Gupta is four times the age of his son. After ten years, the age of Mr. Gupta will be only twice the age of his son. Find the present age of Mr. Gupta‘s son. Solution The present age of Mr. Gupta‘s son = (n 2 -1/n 1 -n 2 )t =(2-1/4-2)10 [Here, n 1 = 4, n 2 = 2 and t=10]= 5years. 3. The age of A, t 1 years ago, was n 1 times the age of B. If t 2 years hence A‘s age would be n 2 times that of B, then, A‘s present age =n 1 (t 1 +t 2 )(n 2 -1)/n 1 -n 2 +t 1 years and, B‘s present age =t 2 (n 2 -1)t 1 (n 1 -1)/n 1 -n 2 years. Explanation Let A‘s present age=x years and B‘s present age=y years. Given:x-t 1 = n 1 (y-t 1 ) and x+t 2 =n 2 (y+t 2 ) i.e. x-n 1 y=(1-n 1 )t 1 .... (1) and x-n 2 y=(-1+n 1 )t 2... (2) Solving (1) and (2), we get x=n 1 (t 1 +t 2 )(n 2 -1)/n 1 -n 2 +t 1 and, y= t 2 (n 2 -1)+t 2 (n 2 -1)/n 1 -n 2. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 58 Illustration 3 10 years ago Anu‘s mother was 4 times older than her daughter. After 10 years, the mother will be twice older than the daughter. Find the present age of Anu. Solution Present age of Anu = t 1 (n 2 -1)+t 1 (n 1 -1)/n 1 -n 2 [Here, n 1 =4, n 2 = 2, t 1 = 10 and t 2 =10] =10(2-1) +10(4-1)/4-2 = 10-30/2 =20years. 4. The sum of present ages of A and B is S year. If, t years ago of A was n times the age of B, then Present age of A =Sn-t(n-1)/n-1 years, and Present age of B= S+t(n-1)/n+1 years. Explanation Let the present ages of A and B be x and y years respectively. Given: x+y = S …(1) and, x-t =n(y-t) or x-ny = (1-n)t Solving (1) and (2), we get x=Sn-t(n-1)/n+1. y= S+t(n-1)/n+1. Illustration 4 The sum of the ages of A and B is 42 years 3 years back, the age of A was 5 times the age of B. Find the difference between the present ages of A and B. Solution Here, S=42, n=5 and t =3 Present age of A = Sn-t(n-1)/n+1= 42×5-3(5-1)/5+1 = 198/6=33 years and, present age of B 5+t(n+1)/n+1 = 42+3(5-1/5+1 =54/6=9years. Difference between the present ages of A and B =33-9=24 years. Note: If, instead of sum(S), difference (D) of their ages is given, replace S by D and in the age denominator (n+1) by (n-1) in the above formula. 5. The sum of present ages of A and B is S years. If, t years hence, the age of A would be n times the age of B, then Present age of A=Sn+t(n-1)/n+1 years and, present age of B = S-t(n-1)/n+1 years. Explanation Let the present ages of A and be x and y years, respectively Given: x+y=S ….(1) and, x+t=n(y+t) or, x-ny=t(n-1) ….(2) Solving(1) and (2), we get x= Sn+t(n-1)/n+1 and, y= S-t(n-1)/n+1 Illustration 5 The sum of the ages of a son and father is 56 years. After four years, the age of the father will be three times that of the son. Find their respective ages. Solution The age of father = Sn+t(n-1)/n+1= 56×3+4(3-1)/3+1 [Here, S=56, t=4 and n=3] = 176/4= 44years. The age of son = Sn-t(n-1)/n+1 = 56-4(3-1)/3+1 = 48/4 = 12 years. 6. If the ratio of the present ages of A and B is a: b and t years hence, it will be c: d, then A‘s present age = at(c-d)/ad-bc and, B‘s present age= bt(c-d)/ ad-bc From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 59 Illustration 6 The ratio of the age of father and son at present is 6 : 1. After 5 years, the ratio will become 7:2. Find the present age of the son. Solution The present age of the son = bt(c-d)/ ad –bc [Here, a =6, b=1, c=7, d=2 and t=5] =1×5(7-2)/6×2-1×7 = 5years. Note: If, with the ratio of present ages, the ratio of ages t years age is given, then replace t by(-t) in the above formula. Illustration 7 6 years ago Mahesh was twice as old as Suresh. If the ratio of their present ages is 9 : 5 respectively, what is the difference between their present ages? Solution Present age Mahesh =--at(c-d)/ad-bc =-9×6(2-1)/1×9-5×2 [Here, a=9, b=5, c=2, d=1 and t =6] =54 years Present age of Suresh -bt(c-d)/ad-bc = -5×6(2-1)/1×9-5×2 =30 years. Difference of their ages =54-30=24 years. MULTIPLE CHOICE QUSTIONS In each of the following questions a number of possible answers are given, out of which one answer is correct. Find out the correct answer. EXERCISE 1. ten years ago, Mohan was thrice as old as Ram was but 10 years hence, he will be only twice as old. Find Mohan‘s present age. a) 60 years b) 80 years c) 70 years d) 76 years 2. The ages of Ram and Shyam differ by 16 years. Six years ago, Mohan‘s age was thrice as that of Ram‘s, find their present ages. a) 14 years, 30 years b) 12 years, 28 years c) 16 years, 34 years d) 18 years, 38 years 3. 15 years hence, Rohit will be just four times as old as he was 15 years ago. How old is Rohit at present? a) 20 b) 25 c) 30 d) 35 4. A man‘s age is 125% of what it was 10 years ago, but 83 1/3 % of what it will be after ten 10 years. What is his present age? a) 45 years b) 50 years c) 55 years d) 60 years 5. If twice the son‘s age in years be added to the father‘s age, the sum is 70 and if twice the father‘s age is added to the son‘s age, the sum is 95. Father‘s age is a) 40 years b) 35 years c) 42 years d) 45years From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 60 6. Three years ago, the average age of a family of 5 members was 17. A baby having been born the average age of the family is the same today? What is the age of the child? a) 3 years b) 5 years c) 2years d) 1 year 7. The ratio of A‘s and B‘s ages is 4:5 If the difference between the present age of A and the age of B 5 years hence is 3, then what is the total of present ages of A and B? a) 68 years b) 72 years c) 76 years d) 64 years 8. The ages of A and B are in the ratio of 6:5 and sum of their ages is 44 years. The ratio of their ages after 8 years will be a) 4 : 5 b) 3 : 4 c) 3 : 7 d) 8 : 7 9. 5 years ago, the combined age of my mother and mine was 40 years. Now, the ratio of our age is 4:1. How old is my mother? (A) 10 (B) 40 (C) 60 (D) 20 (E) 50 10. Honey was twice as old as Vani 10 years ago. How old is Vani today if Honey will be 40 years old 10 years hence? a) 20 b) 25 c) 15 d) 35 e) 30 11. One year ago, a mother was 4 times older to her son. After 6 years, her age become more than double her son‘s age by 5 years. The present ratio of their age will be? a) 13 : 12 b) 11 : 13 c) 3 : 1 d) 25 : 7 e) 4 : 3 12. Vandana‘s mother is twice as old as her brother. She is 5 years younger to her brother but 3 years older to her sister. If her sister is 12 years of age, how old is her mother? a) 30 b) 35 c) 45 d) 40 e) 50 13. Sonu is 4 years younger Manu while Dolly is four years younger to Sumit but 1/5 times as old as Sonu. If Sumit is eight years old, how many times as old is Manu as Dolly? a) 3 b) ½ c) 2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 61 d) 1 e) ¼ 14. Our mother is 3 times as old as my brother and I am 1/3 rd times older than my brother. If 4 years ago I was as old as my brother today, what is the age of my mother. a) 40 b) 36 c) 44 d) 42 e) 48 15. Ruchi‘s age was double that of Niti 2 years ago. If Ruchi was 2 years older to Niti then, try to guess how old she is today. a) 6 b) 4 c) 8 d) 2 e) 20 16. If we add the age of three brothers Sunil, Sanjay and Sonu, then it becomes 60 years today. If 6 years ago the Sonu was of half the age of Sanjay and 1/3 rd to the age of Sunil, then find out the present age of Sanjay. a) 14 b) 15 c) 16 d) 18 e) 24 17. Sonu‘s age is 2/3 rd of Manu‘s. After 5 years Sonu will be 45 years old. Manu‘s present age is a) 55 b) 56 c) 58 d) 60 e) 64 18. Ratio of Sonu‘s age to Manu‘s is equal to 4:3. If Sonu will be 26 years old after 6 years, the present age of Manu is a) 11 b) 15 c) 14 d) 17 e) 13 19. Binny is born on 1 st October. He is younger to Sunny by one week and two days. If on 1 st October it was a Saturday, then Sunny‘s birthday will come on which day this year? (A) Wednesday (B) Thursday (C) Monday (D) Saturday (E) Sunday 20. Binny is half as old as Sunny. Chinky is twice old as Sunny. How many times is Chinky as old as Binny? (A) 6 (B) 4 (C) 8 (D) 3 (E) 2 21. My age becomes half that of my brother‘s if we simply add 2 years to his present age. If I am 25 years old today, my brother will be a) 46 b) 48 c) 44 d) 36 e) 38 22. My age is 2 years less than twice that of my brother. If I am sixteen years old, how old is my brother? a) 3 b) 18 c) 9 d) 27 e) 6 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 62 PLANE GEOMETRY Lines And Angles Any two straight lines which meet at a point make an angle. The angle made by the two straight lines could be any of the following. Some basic properties of angles. Any straight line makes an angle of 180 0 . In the figure 1 and 2 are called adjacent angles. The sum of these angles is equal to 180 0 . They are called supplementary angles to each other. The sum of the angles made at a point is equal to 360 0 . So, (1+2+3+4+5) = 360 0 . Whe n two lines intersect as in the adjacent figure they form a pair of vertically opposite angle (1, 4) and (2, 3). A pair of vertically opposite angles are equal. So, 1 = 4 and 2 = 3. Parallel lines AB and CD are lines that are separated by a constant distance. They do not have any point of intersection. Any line that cuts a pair of parallel lines is called a transversal. The angles formed by the transversal with the parallel lines have the following properties. (a) The correcponding angles are equal i.e. 1 = 5, 2 = 6, 3 = 7, and 4 = 8. (b) The alternate angles are equal 4 = 5, 3 = 6. (c) The interior angles add up to 180 0 , i.e. 4 + 6 = 180 0 and 3 + 5 = 180 0 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 63 Conversely, whenever the corresponding angles are equal or the alternate angles are equal or the interior angles are supplementary, when a straight line cuts two other lines, then we can conclude that the two lines are parallel. If the sum of two angles is 90 0 , then they are complementary to each other. If they add up to 180 0 , then they are said to be supplementary to each other. If more than two straight lines intersect at one and the same point they are called concurrent lines. If there are three lines, if no two of them are parallel and they are not concurrent, then they form a closed figure. Such closed figures formed by three lines are called triangles. Definition and Basic Properties A triangle is a plane figure bounded by three straight lines.  In a triangle, the side which is opposite to (or facing) the largest angle is the longest side and the side which is facing the smallest angle is the shortest side.  The sum of the lengths of two sides of a triangle is always greater than the length of the third side.  The sum of the internal angles in a triangle is equal to 180 0 . Nomenclature Associated The corners of the triangle are called its vertices. Generally, the side opposite a vertex is represented by the same nomenclature but in a different case. For example, the side opposite ―ZA‖ would be named ―a‖. The side opposite ―ZB‖ and ―ZC‖ would be named as ―b‖ and ―c‖ respectively. The angles associated with these vertices are called the interior angles. Each interior has an associated exterior angle which can be obtained by extending any one side of the angle. The interior and the exterior angels are supplementary. Altitude or Height The perpendicular dropped to the side of a triangle from the vertex opposite that side. The perimeter of any triangle is the sum of the lengths of its sides. Perimeter = a + b + c Semi – perimeter (s) = (a + b + c)/2 Area = (base × height)/2 (OR) ) )( )( ( c s b s a s s ÷ ÷ ÷ Types of Triangles Scalene, when its sides (and angles) are unequal Perimeter = a + b + c From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 64 Area = ) )( )( ( c s b s a s s ÷ ÷ ÷ Equilateral, when all its sides (and angles) are equal. AB=AC=BC ¬ a=b=c. ZA=ZB=ZC=60 0 . Perimeter = 3a Area = 2 4 3 a Height = a 4 3 Isosceles, when two of its sides (and two angles opposite the two equal sides) are equal. AB = AC and ZB = ZC. Perimeter = 2a + b Area = 2 2 4 2 1 b a b ÷ × × Height = 2 2 4 2 1 b a ÷ × Right Angled, when one of its angles is a right angle. ZB = 90 0 and ZA+ZC = 90 0 Perimeter = a + b + c Area = ac 2 1 Pythagoras Theorem The side opposite the right angle is called the hypotenuse. Then, from Pythagoras theorem, a 2 +c 2 = b 2 A triplet is a set of numbers which will satisfy the Pythagoras theorem. The frequently used triplets are (3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17) (9, 40, 41) (11, 60, 61) (12, 35, 37) (16, 63, 65) (20, 21, 29). The multiples of triplets are also triplets. Example : 6, 8, 10. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 65 Right Angled Isosceles, when one of its angles is a right angle, and the sides containing the right angle are equal. AB = BC and ZA = ZC = 45 0 Perimeter = 2a + b Area = 2 2 1 a = 2 4 1 b Simple Trigonometric Ratios Consider a right triangle : With reference to angle A, the following trigonometric ratios are defined: Sine of the angle : sin (A) or sinA = (Opposite side/Hypotenuse) Cosine of the angle : cos(A) or cosA = (Adjacent side/Hypotenuse) Tangent of the angle : tan(A) or tanA = (Opposite side/Adjacent side) A quadrilateral is a polygon with four sides. A quadrilateral has four sides and four internal angles. The sum of the internal angles, i.e. ZA + ZB + ZC + ZD = 360 0 , since the quadrilateral can be split into two triangles. Quadrilaterals can be classified based on relationships within its sides. Parallelogram A quadrilateral in which the opposite sides are parallel is called a parallelogram. Basic Properties  The opposite sides are parallel and of equal length. AB=DC and AD = BC.  The sum of any two adjacent interior angles is equal to two right angles or 180 0 . ZA+ZB=+ZB+ZC=ZC+ZD=ZD+ZA=180 0 .  The OPPOSITE Angles are equal in magnitude. ZA=ZC and ZB=ZD.  The diagonals of a parallelogram are not equal in magnitude, but they bisect each other, and from two pairs of congruent triangles.  Perimeter = (Twice the sum of non parallel sides) = 2(AB + BC) From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 66  Area = (base × height) = (b × h) where the height h is the perpendicular distance between the base and the side parallel to it. If D is one of the diagonals, and if a, b are two adjacent sides of parallelogram, then Area | | 2 , ) )( )( ( 2 c b a s where D s b s a s s + + = ÷ ÷ ÷ × =  BD 2 +AC 2 = 2(BC 2 + CD 2 )  The line joining the midpoints of two adjacent sides of a parallelogram is parallel and half the length of the corresponding diagonal of the parallelogram. This line cuts the other diagonal in the ratio of 1:3.  The line joining the midpoint of a side of a parallelogram with one of the opposite vertices cuts one of the diagonals in the ratio of 1:2. Rhombus A rhombus is a special case of a parallelogram where all the sides are of equal length. Basic Properties ¬ The opposite sides are parallel and all sides are of equal length. AB = BC = CD = DA. ¬ The sum of any two adjacent interior angles is equal to two right angles or 180 0 . ZA+ZB=ZB+ZC=ZC+ZD=ZD+ZA=180 0 . ¬ The opposite angles are equal. ZA=ZC and ZB=ZD. ¬ The diagonals bisect each other at right angles and form four right angled triangles. Thus BE=DE=BD/2 and AE=CE=AC/2 and ZAEB=ZBEC=ZCED=ZDEA=90 0 . ¬ Perimeter = (4×side) = 4AB = 4BC = 4CD = 4AD ¬ The area of a rhombus = half the product of its diagonals = ½ m× AC ×BD. ¬ Areas of the four right triangles, AAEB, ABEC, ACED, ADEA are equal and each equals 1/4 th the area of the Rhombus. ¬ 2 2 2 2 2 ) ( | . | \ | + | . | \ | = BD AC side Rectangle A rectangle is a special case of a parallelogram where the adjacent sides are perpendicular to each other. Basic Properties 1. The opposite sides are parallel and of equal length. AB=CD and AD=BC. The longer side is called the length (L) and the shorter side the Breadth (B). From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 67 2. The adjacent sides are perpendicular. ZA=ZB=ZC=ZD=90 0 3. The diagonals of a rectangle are of equal length and bisect each other. AC=BD; AE=BE=CE=DE. 4. Perimeter = 2(L + B) Area = (Length × Breadth) = L × B 5. If L is the length and D the diagonal, from Pythagoras theorem, Breadth 2 2 L D B ÷ = Conversely, if B is the breadth and D the diagonal, from Pythagoras theorem, Length 2 2 B D L ÷ = Parallel Path (Shaded Portion) ABCD is a rectangle. SU and TV are two paths drawn parallel to the W is the width of each parallel path. Area of two parallel paths (shaded portion) = W(l+b-W) Square A square is a special case of a parallelogram where all the sides are of equal length and perpendicular to each other. Thus it is a rhombus and a rectangle also. Basic Properties ¬ All the sides of the square are equal. Opposite sides are parallel and adjacent sides are perpendicular. AB=BC=CD=DA. ZA=ZB=ZC=ZD=90 0 . ¬ The diagonals of a square are of equal length and bisect each other at right angles, AC=BD and AE=BE=CE=DE. ¬ Perimeter = 4a Area = a 2 ¬ Diagonal of the square, from Pythagoras theorem = ( ) 2 a Thus area of square = (Diagonal) 2 /2 ¬ Of all quadrilateral with a given area, the square is the one which has the least perimeter. ¬ Of all quadrilaterals with a given perimeter, the square is the one which has the greatest area. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 68 Trapezium A trapezium is a quadrilateral where only one pair of opposite sides are parallel. All parallelograms are thus trapeziums (converse is not true). The Area of a trapezium = half the sum of the lengths of the parallel sides multiplied by the perpendicular distance between them. Area = h BC AD ) ( 2 1 + Circles A circle is a set of points which are equidistant from a given point. The given point is known as the center of that circle. The angle in a circle is 360 0 . Basic Constructs The distance from the centre of the circle to any point on it is known as the radius (R). A circle is completely defined by its radius and its position can be fixed if its centre‘s position is given. Twice the radius is known as the diameter (D). Thus D = 2R. Circumference All the points which lie on the circle constitute the circumference. The ratio of the circumference to the diameter is a constant for any circle and is given by HD=2HR. Secant Any line which passes through the circle is called a secant. A secant cuts the circumference of the circle at two points. Chord Any line segment whose ends lie on the circumference of the circ le is called a chord of that circle. A chord which passes through the centre of the circle is the diameter. Tangent A line which touches the circle at one point is called a tangent to that circle. The point common to the tangent and the circumference of the circle is called the point of contact. The radius of the circle and the tangent to the circle are perpendicular at the point of contact. Area From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 69 The Area contained the circle is determined by HR 2 , where R is the radius of the circle. Arc A part of the circumference of a circle is called an arc. Each arc has an angle associated with it which it subtends at the centre of the circle. This central angle is related to the length of the arc through the metric of radian. By definition, the radian is that angle which is subtended at the centre of a circle by an arc of length equal to the radius of that circle. Consequently, since the full circle is 2Hr (circumference of circle), a full circle would correspond to 2H radians. There is a direct relationship between degrees and radians i.e. 360 0 is 2H radians. Similarly, 180 0 would be Hradians and so on. Sector A sector is part cut from the circle bounded by an arc and the radii drawn from the centre of the circle to the arc‘s ends. The radii form the centrally subtended angle between them. The area of a sector is directly proportional to this angle and it is equal to that of a full circle if this angle is 360 0 . Arcs and Sectors For a circle of radius R, if the central angle subtended by an arc is o 0 , then Length of Arc | | . | \ | = 0 0 360 2 o tR L Perimeter of sector = ( ¸ ( ¸ | | . | \ | + 0 0 360 2 2 o tR R Area of the sector = | | . | \ | 0 0 360 2 o tR From the above two relations, it is obvious that | . | \ | = | | . | \ | R Sector of Area Arc the of Length 2 In case the central angle is given in radians (say | radians), then Length of arc = R| Perimeter of the sector = (2R) + (R|) Area of the sector = ½ (R 2 |) Circular Pathway OAC is a circle of radius = r, there is pathway, outside the circle of width = W Area of circular pathway = t × W(2r + W) From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 70 When, the pathway is inside the circle, Area of circular pathway = t × W(2r – W) Properties of Circles The properties of circles can be categorized in the following classes: A: Arcs, Chords and Central angles B: Angles in a circle C: Chords in a circle D: Tangents to a circle E: Pair of circles F: Cyclic quadrilaterals A. ARCS, Chords And Central Angles  In equal circles (or in the same circle), if two arcs are equal, the chords associated with the arcs are equal.  In equal circles (or in the same circle), if two arcs subtend equal angles at the centres or at the circumferences of the circles, then they are equal.  In equal circles (or in the same circle), if two chords are equal, then the arcs which they cut off are equal. B. Angle in a circle  The angle which an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference.  Angles in the same segment of a circle are equal.  The angle in a semicircle is a right angle. C. Chords in a circle  A straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is at right angles to the chord, i.e. if OP bisects AB then OP ± AB. Conversely, the perpendicular to chord from the centre bisects the chord, i.e. if OP±AB then AP = PB.  Equal chords of a circle are equidistant from the centre. Conversely, The chords that are equidistant from the centre are equal.  If two chords of a circle, AB & CD, intersect internally at O, then AO × OB = CO × OD. D. Tangents to a circle From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 71  The tangent at any pint of a circle and the radius through that point of contact are perpendicular to each other. OT is perpendicular to PT.  If two tangents are drawn to a circle from an outside point, the length of the tangents from the external point to their respective points of contact are equal, i.e. PA = PB.  The angle which a chord makes with a tangent at its point of contact is equal to any angle in the alternate segment, i.e. ZPTA = ZTBA.  If PT is a tangent (with P being an external point and T being the point of contact) and PAB is a secant to circle (with A and B as the points where the secant cuts the circle), then PT2 = PA × PB E. Pair of Circles  If two circles touch each other, the point of contact of he two circles lies on the straight line through the centres of the circles, i.e. the points A, C, B are collinear.  In a given pair of circles, there are two types of tangents – the direct tangents and the cross (or transverse) tangents. In the figure given alongside, the direct tangents are AB and CD while EF and GH are the transverse tangents.  When two circles of radii R 1 and R 2 have their centres at a distance of d, The length of direct common tangent = 2 2 r d ÷ , where r = R 1 – R 2 The length of the transverse tangent is 2 2 1 2 ) ( R R d + ÷ Note that if the two circles touch d = R 1 + R 2 F. Cyclic Quadrilateral From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 72 A quadrilateral whose vertices lie on the circumference of a circle | called cyclic quadrilateral.  The opposite angles of a cyclic quadrilateral are supplementary. In the given figure ZADC + ZABC = o + | = 180 0 .  The area of a cyclic quadrilateral with sides a, b, c, d is ) )( )( )( ( d s c s b s a s ÷ ÷ ÷ ÷ , where s = 2 d c b a + + + Geometrical relationships  Of all surface with a given area, the circle is the one which has the least perimeter.  Of all surfaces with a given perimeter, the circle is the one which has the greatest area.  If a circle is inscribed in a square, the diameter of the circle is equal to the side of the square. The area of the largest circle that can be inscribed in a square of side ‗a‘ is 2 4 a | . | \ |t  If a square/rectangle is inscribed in a circle, then the diagonal of the square/rectangle is equal to the diameter of the circle. Area of a square inscribed in a circle of radius r is 2 r 2 .  If a circle is inscribed in a rectangle, the diameter of the circle is equal to the smaller side of the rectangle.  The area of a circle circumscribing an equilateral triangle of side ‗a‘ is . 3 2 a | . | \ |t The area of a circle inscribed in an equilateral triangle of side ‗a‘ is 2 12 a | . | \ | t .  Two circles are said to be concentric if their centres coincide. Solids Solids can be classified into two main divisions viz. solids with flat surfaces and solids with curved surface. Solids with flat surfaces are described as polyhedrons while surfaces of revolution would form the important part of solids with curved surfaces. Polyhedrons A polyhedron is a closed solid object formed using planar surfaces. It has an overall convex shape, no curved surfaces and has no perforations. Cuboid A rectangular solid having six faces – all of which are rectangles. Cuboid metrics From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 73 Let the cuboid have dimensions of length L, breadth B, and height H. Volume = (cross section area × height) = L × B × H cubic units. Area of four walls (excluding top and bottom faces) = 2[(L+B)H]. Total surface area = 2(LB+BH+HL) sq. units. Length of the diagonals of a cuboid = 2 2 2 H B L + + units Hollow cuboid Consider for example a carton used for packing. It is not completely solid. Besides, L, B and H, there will another dimension, which is the thickness T. If L, B and H are the external dimensions, then the internal dimensions are [L-2T], [B-2T] and [H-2T]. Volume of material used = (External Volume – internal Volume). Thus Volume of material used = LBH – [(L-2T)(B-2T)(H-2T)] cubic units. However, if it is an open box, that is the top face is missing, then the internal dimensions will be [L-2T], [B-2T] and [H-T]. Cube A rectangular 6-faced solid whose every face is a square. Every cube is a cuboid also, such that L = B = H. Cube Metrics Let the length of the edge of the cube be A. Volume = [(area of cross section) × (height)] = A 3 cubic units. Area of four walls of a cube (excluding the top and bottom faces) = 4A2. Total surface area of a cube = (sum of areas of all six faces) = 6A 2 sq. units Length of a diagonals of a cube = A\3 units. Hollow cube If the thickness is T, Volume of material used = External Volume – internal Volume = A 3 -{A-2T} 3 , in the case of closed box. = A 3 – [{A-2T} 2 {A-T}], in the case of an open box. Pyramid From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 74 A pyramid is a polyhedron with a polygonal base and planar angular surfaces on the side leading to an apex point or a vertex at the top. A regular pyramid has a regular polygonal base. A right regular pyramid is a regular pyramid in which all the side surfaces (all surfaces except the base) are equal. The axis of a pyramid is an imaginary line joining the midpoint of the base polygon to the top vertex of the pyramid. In a right regular pyramid, the axis is perfectly vertical (i.e. it is perpendicular to the base). In case the axis is not perpendicular, a tilted pyramid is formed. Pyramid metrics (Right regular pyramid with base polygon of N sides) Volume = 1/3 × Area of the base × height , where the height is the length of the axis. Surface area = Area of the base + (N × area of each side) Prism Prism contains similar top and bottom face and the side faces are rectangular shape. Let P is the perimeter of base, H is the height of the prism and B is the base area of the prism then volume = B × H, Lateral surface are = P × H, total surface area = (P × H) + 2 × B. Solids with curved surfaces These are generally obtained by revolving a planar surface about some axis. Symmetrical curved solids of revolution are obtained when the surface being revolved is symmetrical (say a regular polygon) and the axis of revolution is also properly chosen. Cylinder A solid formed when a rectangle is revolved about one of its sides is called the right circular cylinder. Cylinder metrics Let the base radius be R and the height be H. Volume of the cylinder = area of cross-section × height or V = tR 2 H cubic units. Curved surface Area of the cylinder (excludes the areas of the top and bottom circular regions) = area of rectangle whose sides are 2HR and H or CSA=2tRH sq. units. Total surface Area = Curved Surface Area + Areas of the top and bottom circular regions or TSA = 2tRH + 2tR 2 =2tR[R+H] sq. units. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 75 If it is a hollow cylinder of thickness T, then internal base radius = r = R – T, Volume of material in a hollow cylinder = External Volume – Internal Volume or V = t(R 2 – r 2 )H cubic units. Cone A solid formed by rotating a right angled triangle about one of the sides containing the right angle. Cone Metrics Let the Base Radius = R, Vertical Height = H, and Slant Height = L. Volume = 1/3 (tR 2 H cubic units) Slant Height L = 2 2 H R + units Curved Surface Area = tRL sq. units total Surface Area = tR(R + L) sq. units Frustrum If a cone is cut by a plane parallel to the base, then the lower part is called the frustrum of the cone, Let the radius of the top = r and the radius of the base – R and height = h. Slant Height L = | | 2 2 r R h ÷ + . Curved Surface Area = t[r + R]L sq. units. Total surface area = t[(r + R)L + r 2 + R 2 ]sq. units. Volume = t | | rR R r h + + | . | \ | 2 2 3 cubic units. Sphere A solid formed when a circle is revolved about its diameter. Sphere Metrics Let the Radius of the Sphere = R. Volume = 4/3 (tR 3 cubic units) Surface Area = 4tR 2 sq. units If R and r are the external and internal radii of a spherical shell, then its Volume = | | 3 3 3 4 r R ÷ cubic units. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 76 Geometrical Relationships  Of all solids with a given volume, the sphere is the one which has the lest surface area.  Of all solids with a given surface area, the sphere is the one which has the greatest volume. Hemisphere Volume = 3 3 2 R t cubic units Surface Area = 3tR 2 sq. units EXERCISE 1. A vertical stick 20 m long casts a shadow 10 m long on the ground. At the same time, a tower casts the shadow 50 m long on the ground. Find the height of the tower. (A) 100 m (B) 120 m (C) 25 m (D) 200 m 2. In the figure, AABC is similar to AEDC. If we have AB = 4 cm, ED = 3 cm, CE = 4.2 and CD = 4.8 cm, find the value of CA and CB (A) 6 cm, 6.4 cm (B) 4.8 cm, 6.4 cm (C) 5.4 cm, 6.4 cm (D) 5.6 cm, 6.4 cm 3. The area of similar triangles, ABC and DEF are 144 cm 2 and 81 cm 2 respectively. If the longest side of larger AABC be 36 cm, then the longest side of smaller ADEF is (A) 20 cm (B) 26 cm (C) 27 cm (D) 30 cm 4. Two isosceles As have equal angles and their areas are in ratio 16 : 25. Find the ratio of their corresponding heights. (A) 4/5 (B) 5/4 (C) 3/2 (D) 5/7 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 77 5. The areas of two similar As are respectively 9 cm 2 and 16 cm 2 . Find the ratio of their corresponding sides. (A) 3 : 4 (B) 4 : 3 (C) 2 : 3 (D) 4 : 5 6. Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, find the distance between their tops. (A) 12 cm (B) 14 cm (C) 13 cm (D) 11 cm 7. The radius of a circle is 9 cm and length of one of its chords is 14 cm. Find the distance of the chord from the center. (A) 5.6 cm (B) 6.3 cm (C) 4 cm (D) 7 cm 8. Find the length of a chord that is at a distance of 12 cm from the center of a circle of radius 13 cm. (A) 9 cm (B) 10 cm (C) 12 cm (D) 5 cm 9. If O is the center of circle, find Zx (A) 35 0 (B) 30 0 (C) 39 0 (D) 40 0 10. Find the value of Zx in the given figure. (A) 120 0 (B) 130 0 (C) 100 0 (D) 150 0 11. Find the value of x in the figure, if it is given that AC and BD are diameters the circle. (A) 60 0 (B) 45 0 (C) 15 0 (D) 30 0 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 78 12. Find the value of x in the given figure. (A) 2.2 cm (B) 1.6 cm (C) 3 cm (D) 2.6 cm 13. Find the value of x in the given figure. (A) 16 cm (B) 9 cm (C) 12 cm (D) 7 cm 14. Find the value of x in the given figure. (A) 13 cm (B) 12 cm (C) 16 cm (D) 15 cm 15. ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. A circle with center O and radius x has been inscribed in AABC. What is the value of x. (A) 2.4 cm (B) 2 cm (C) 3.6 cm (D) 4 cm 16. In the given figure ZAB, is the diameter of the circle and ZPAB = 250. Find ZTPA. (A) 50 0 (B) 65 0 (C) 70 0 (D) 45 0 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 79 17. In the given figure find ZADB. (A) 132 0 (B) 144 0 (C) 48 0 (D) 96 0 18. In the given two straight line, PQ and RS intersect each other at O. If ZSOT = 75 0 , find the value of a, b and c. (A) a = 84 0 , b = 21 0 , c = 48 0 (B) a = 48 0 , b = 20 0 , c = 50 0 (C) a = 72 0 , b = 24 0 , c = 54 0 (D) a = 64 0 , b = 28 0 , c = 45 0 19. In the following figure A, B, C and D are the concyclic points. Find the value of x. (A) 130 0 (B) 50 0 (C) 60 0 (D) 30 0 20. In the following figure, it is given that O is the center of the circle and ZAOC = 140 0 . Find ZABC. (A) 110 0 (B) 120 0 (C) 115 0 (D) 130 0 21. In the following figure, O is the center of the circle and ZABO = 30 0 , find ZACB. (A) 60 0 (B) 120 0 (C) 75 0 (D) 90 0 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 80 22. In the following figure, find the value of x (A) 40 0 (B) 25 0 (C) 30 0 (D) 45 0 23. If L 1 ||L 2 in the figure below, what is the value of x. (A) 80 0 (B) 100 0 (C) 40 0 (D) Cannot be determined 24. Find the perimeter of the given figure. (A) (32 + 3t) cm (B) (36 + 6t) cm (C) (46 + 3t) cm (D) (26 + 3t) cm 25. In the figure, AB is parallel to CD and RD SL TM AN , and BR : RS : ST : TA = 3 : 5 : 2 : 7. If it is known that CN = 1.333 BR. Find the ratio of BF : FG : GH : HI : IC (A) 3 : 7 : 2 : 5 : 4 (B) 3 : 5 : 2 : 7 : 4 (C) 4 : 7 : 2 : 5 : 3 (D) 4 : 5 : 2 : 7 : 3 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 81 MENSURATION Solids Solids can be classified into two main divisions viz. solids with flat surfaces and solids with curved surface. Solids with flat surfaces are described as polyhedrons while surfaces of revolution would form the important part of solids with curved surfaces. Polyhedrons A polyhedron is a closed solid object formed using planar surfaces. It has an overall convex shape, no curved surfaces and has no perforations. Cuboid A rectangular solid having six faces – all of which are rectangles. Cuboid metrics Let the cuboid have dimensions of length L, breadth B, and height H. Volume = (cross section area × height) = L × B × H cubic units. Area of four walls (excluding top and bottom faces) = 2[(L+B)H]. Total surface area = 2(LB+BH+HL) sq. units. Length of the diagonals of a cuboid = 2 2 2 H B L + + units Hollow cuboid Consider for example a carton used for packing. It is not completely solid. Besides, L, B and H, there will another dimension, which is the thickness T. If L, B and H are the external dimensions, then the internal dimensions are [L-2T], [B-2T] and [H-2T]. Volume of material used = (External Volume – internal Volume). Thus Volume of material used = LBH – [(L-2T)(B-2T)(H-2T)] cubic units. However, if it is an open box, that is the top face is missing, then the internal dimensions will be [L-2T], [B-2T] and [H-T]. Cube A rectangular 6-faced solid whose every face is a square. Every cube is a cuboid also, such that L = B = H. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 82 Cube Metrics Let the length of the edge of the cube be A. Volume = [(area of cross section) × (height)] = A 3 cubic units. Area of four walls of a cube (excluding the top and bottom faces) = 4A2. Total surface area of a cube = (sum of areas of all six faces) = 6A 2 sq. units Length of a diagonals of a cube = A\3 units. Hollow cube If the thickness is T, Volume of material used = External Volume – internal Volume = A 3 -{A-2T} 3 , in the case of closed box. = A 3 – [{A-2T} 2 {A-T}], in the case of an open box. Pyramid A pyramid is a polyhedron with a polygonal base and planar angular surfaces on the side leading to an apex point or a vertex at the top. A regular pyramid has a regular polygonal base. A right regular pyramid is a regular pyramid in which all the side surfaces (all surfaces except the base) are equal. The axis of a pyramid is an imaginary line joining the midpoint of the base polygon to the top vertex of the pyramid. In a right regular pyramid, the axis is perfectly vertical (i.e. it is perpendicular to the base). In case the axis is not perpendicular, a tilted pyramid is formed. Pyramid metrics (Right regular pyramid with base polygon of N sides) Volume = 1/3 × Area of the base × height , where the height is the length of the axis. Surface area = Area of the base + (N × area of each side) Prism Prism contains similar top and bottom face and the side faces are rectangular shape. Let P is the perimeter of base, H is the height of the prism and B is the base area of the prism then volume = B × H, Lateral surface are = P × H, total surface area = (P × H) + 2 × B. Solids with curved surfaces These are generally obtained by revolving a planar surface about some axis. Symmetrical curved solids of revolution are obtained when the surface being revolved is symmetrical (say a regular polygon) and the axis of revolution is also properly chosen. Cylinder A solid formed when a rectangle is revolved about one of its sides is called the right circular cylinder. Cylinder metrics From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 83 Let the base radius be R and the height be H. Volume of the cylinder = area of cross-section × height or V = tR 2 H cubic units. Curved surface Area of the cylinder (excludes the areas of the top and bottom circular regions) = area of rectangle whose sides are 2HR and H or CSA=2tRH sq. units. Total surface Area = Curved Surface Area + Areas of the top and bottom circular regions or TSA = 2tRH + 2tR 2 =2tR[R+H] sq. units. If it is a hollow cylinder of thickness T, then internal base radius = r = R – T, Volume of material in a hollow cylinder = External Volume – Internal Volume or V = t(R 2 – r 2 )H cubic units. Cone A solid formed by rotating a right angled triangle about one of the sides containing the right angle. Cone Metrics Let the Base Radius = R, Vertical Height = H, and Slant Height = L. Volume = 1/3 (tR 2 H cubic units) Slant Height L = 2 2 H R + units Curved Surface Area = tRL sq. units total Surface Area = tR(R + L) sq. units Frustrum If a cone is cut by a plane parallel to the base, then the lower part is called the frustrum of the cone, Let the radius of the top = r and the radius of the base – R and height = h. Slant Height L = | | 2 2 r R h ÷ + . Curved Surface Area = t[r + R]L sq. units. Total surface area = t[(r + R)L + r 2 + R 2 ]sq. units. Volume = t | | rR R r h + + | . | \ | 2 2 3 cubic units. Sphere From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 84 A solid formed when a circle is revolved about its diameter. Sphere Metrics Let the Radius of the Sphere = R. Volume = 4/3 (tR 3 cubic units) Surface Area = 4tR 2 sq. units If R and r are the external and internal radii of a spherical shell, then its Volume = | | 3 3 3 4 r R ÷ cubic units. Geometrical Relationships  Of all solids with a given volume, the sphere is the one which has the lest surface area.  Of all solids with a given surface area, the sphere is the one which has the greatest volume. Hemisphere Volume = 3 3 2 R t cubic units Surface Area = 3tR 2 sq. units EXERCISE 1. In a right angled triangle, find the hypotenuse if base and perpendicular are respectively 36015 cm and 48020 cm. (A) 69125 cm (B) 60025 cm (C) 391025 cm (D) 60125 cm 2. The perimeter of an equilateral triangle is 72\3 cm. Find its height. (A) 63 metres (B) 24 metres (C) 18 metres (D) 36 metres 3. The inner circumference of a circular track is 440 cm. The rack is 14 cm wide. Find the diameter of the outer circle of the track. (A) 84 cm (B) 168 cm (C) 336 cm (D) 77 cm From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 85 4. A race track is in the form of a ring whose inner and outer circumference are 352 metre and 396 metre respectively. Find the width of the track. (A) 7 metres (B) 14 metres (C) 14t metres (D) 7t metres 5. The outer circumference of a circular track is 220 metre. The track is 7 metre wide everywhere. Calculate the cost of leveling the track at the rate of 50 paise per square metre. (A) Rs. 1556.5 (B) Rs. 3113 (C) Rs. 693 (D) Rs. 1386 6. Find the area of a quadrant of a circle whose circumference is 44 cm. (A) 77 cm 2 (B) 38.5 cm 2 (C) 19.25 cm 2 (D) 19.25t cm 2 7. A pit 7.5 metre long, 6 metre wide and 1.5 metre deep is dug in a field. Find the volume of soil removed in cubic metres. (A) 135 m 3 (B) 101.25 m 3 (C) 50.625 m 3 (D) 67.5 m 3 8. Find the length of the longest pole that can be placed in an indoor stadium 24 metre long, 18 metre wide and 16 metre high. (A) 30 metres (B) 25 metres (C) 34 metres (D) \580 metres 9. The length, breadth and height of a room are in the ratio of 3 : 2: 1. If its volume be 1296 m 3 , find its breadth. (A) 12 metres (B) 18 metres (C) 16 metres (D) 24 metres 10. The volume of a cube is 216 cm 3 . Part of this cube is then melted to form a cylinder of length 8 cm. Find the volume of the cylinder. (A) 342 cm 3 (B) 216 cm 3 (C) 36 cm 3 (D) Data inadequate 11. The whole surface of a rectangular block is 8788 square cm. If length, breadth and height are in the ratio of 4:3:2, find length. (A) 26 cm (B) 52 cm (C) 104 cm (D) 13 cm From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 86 12. Three metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted together and formed into a single cube. Find the side of the resulting cube. (A) 11 cm (B) 12 cm (C) 13 cm (D) 24 cm 13. Find curved and total surface area of a conical flask of radius 6 cm and height 8 cm. (A) 60t, 96t (B) 20t, 96t (C) 60t, 48t (D) 30t, 48t 14. Volume of a right circular cone is 100t cm 3 and its height is 12 cm. Find its curved surface area. (A) 130t cm 2 (B) 65t cm 2 (C) 204t cm 2 (D) 65 cm 2 15. The diameters of two cones are equal. If their slant height be in the ratio 5 : 7, find the ratio of their curved surface areas. (A) 25 : 7 (B) 25 : 49 (C) 5 : 49 (D) 5 : 7 16. The curved surface area of a cone is 2376 square cm and its slant height is 18 cm. Find the diameter. (A) 6 cm (B) 18 cm (C) 84 cm (D) 12 cm 17. The ratio of radii of a cylinder to a that of a cone is 1 : 2. If their heights are equal, find the ratio of their volume? (A) 1 : 3 (B) 2 : 3 (C) 3 : 4 (D) 3 : 2 18. A silver wire when bent in the form of a square, encloses an area of 484 cm 2 . Now if the same wire is bent to form a circle, the area of enclosed by it would be (A) 308 cm 2 (B) 196 cm 2 (C) 616 cm 2 (D) 88 cm 2 19. The circumference of a circle exceeds its diameter by 16.8 cm. Find the circumference of the circle. (A) 12.32 cm (B) 49.28 cm (C) 58.64 cm (D) 24.64 cm 20. A bicycle wheel makes 5000 revolutions in moving 11 km. What is the radius of the wheel? From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 87 (A) 70 cm (B) 135 cm (C) 17.5 cm (D) 35 cm 21. The volume of a right circular cone is 100t cm 3 and its height is 12 cm. Find its slant height. (A) 13 cm (B) 16 cm (C) 9 cm (D) 26 cm 22. The short and the long hands of a clock are 4 cm and 6 cm long respectively. What will be sum of distances traveled by their tips in 4 days? (Take t = 3.14) (A) 954.56 cm (B) 3818.24 cm (C) 2909.12 cm (D) None of these 23. The surface areas of two spheres are in the ratio of 1 : 4. Find the ratio of their volumes. (A) 1 : 2 (B) 1 : 8 (C) 1 : 4 (D) 1 : 2 24. The outer and inner diameters of a spherical shell are 10 cm and 9 cm respectively. Find the volume of the metal contained in the shell. (Use t = 22/7) (A) 6956 cm 3 (B) 141.95 cm 3 (C) 283.9 cm 3 (D) 478.3 cm 3 25. The radii of two spheres are in the ratio of 1 : 2. Find the ratio of their surface areas. (A) 1 : 3 (B) 2 : 3 (C) 1 : 4 (D) 3 : 4 26. A sphere of radius r has the same volume as that of a cone with a circular base of radius r. Find the height of cone. (B) 2r (A) r/3 (B) 4r (C) (2/3)r 27. Find the number of bricks, each measuring 25 cm x 12.5 cm x 7.5 cm, required to construct a wall 12 m long, 5 m high and 0.25 m thick, while the sand and cement mixture occupies 5% of the total volume of wall. (A) 6080 (B) 3040 (C) 1520 (D) 12160 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 88 28. A road that is 7 m wide surrounds a circular path whose circumference is 352 m. What will be the area of the road? (A) 2618 cm 2 (B) 654.5 cm 2 (C) 1309 cm 2 (D) 5236 cm 2 29. In a shower, 10 cm of rain falls. What will be the volume of water that falls on 1 hectare area of ground? (A) 500 m 3 (B) 650 m 3 (C) 2250 cm 2 (D) 700 cm 2 30. Seven equal cubes each of side 5 cm are joined end to end. Find the surface area of the resulting cuboid. (A) 750 cm 2 (B) 1500 cm 2 (C) 2250 cm 2 (D) 700 cm 2 31. In a swimming pool measuring 90 m by 40 m, 150 men take a dip. If the average displacement of water by a man is 8 cubic metres, what will be rise in water level? (A) 30 cm (B) 33.33 cm (C) 20 cm (D) 25 cm 32. How many metres of cloth 5 m wide will be required to make a conical tent, the radius of whose base is 7 m and height is 24 m? (A) 55 m (B) 330 m (C) 220 m (D) 110 m 33. Two cones have their heights in the ratio 1 : 2 and the diameters of their bases are in the ratio 2 : 1. What will be the ratio of their volumes? (A) 4 : 1 (B) 2 : 1 (C) 3 : 2 (D) 1 : 1 34. A conical tent is to accommodate 10 persons. Each person must have 6 m 2 space to sit and 30 m 3 of air to breath. What will be the height of the cone? (A) 37.5 m (B) 15 m (C) 75 m (D) None of these 35. A closed wooden box measures externally 10 cm long, 8 cm broad and 6 cm high. Thickness of wood is 0.5 cm. Find the volume of wood used. (A) 230 cubic cm (B) 165 cubic on (C) 330 cubic on (D) None of these From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 89 36. A cuboid of dimension 24 cm x 9 cm x 8 cm is melted and smaller cubes are of side 3 cm is formed. Find how many such cubes can be formed? (A) 27 (B) 64 (C) 54 (D) 32 37. Three cubes each of volume of 216 m 3 are joined end to end. Find the surface area of the resulting figure. (A) 504 m 2 (B) 216 m 2 (C) 432 m 2 (D) None of these 38. A hollow spherical shell is made of a metal of density 4.9 g/cm 3 . If its internal and external radii are 10 cm and 12 cm respectively, find the weight of the shell. (Take t = 3.1416) (A) 5016 gm (B) 1416.8 gm (C) 14942.28 gm (D) 5667.1 gm 40. The largest cone is formed at the base of a cube of side measuring 7 cm. Find the ratio of volume of cone to cube. (A) 20 : 21 (B) 22 : 21 (C) 21 : 22 (D) 11 : 42 41. A spherical cannon ball, 28 cm in diameter, is melted and cast into a right circular conical mould the base of which is 35 cm in diameter. Find the height of the cone correct up to two places of decimals. (A) 8.96 cm (B) 35.84 cm (C) 5.97 cm (D) 17.92 cm 42. Find the area of the circle circumscribed about a square each side of which is 10 cm. (A) 314.28 cm 3 (B) 157.14 cm 3 (C) 150.38 cm 3 (D) 78.57 cm 3 43. Find the radius of the circle inscribed in a triangle whose sides are 8 cm, 15 cm and 17 cm. (A) 4 cm (B) 5 cm (C) 3 cm (D) 2\2 cm From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 90 44. In the given diagram a rope is wound round the outside of a circular drum whose diameter is 70 cm and a bucket is tied to the other end of the rope. Find the number of revolutions made by the drum if the bucket is raised by 11 m. (A) 10 (B) 2.5 (C) 5 (D) 5.5 45. A cube whose edge is 20 cm long has circle on each of its faces painted black. What is the total area of the unpainted surface of the cube if the circles are of the largest area possible? (A) 85.71 cm 2 (B) 257.14 cm 2 (C) 514.28 cm 2 (D) 331.33 cm 2 46. The areas of three adjacent faces of a cuboid are x, y, z. If the volume is V, then V 2 will be equal to (A) xy/z (B) yz/x 2 (C) x 2 y 2 /z 2 (D) xyz 47. In the adjacent figure, find the area of the shaded region. (Use = 22/7) (A) 15.28 cm 2 (B) 61.14 cm 2 (C) 30.57 cm 2 (D) 40.76 cm 2 48. The diagram represents the area swept by the wiper of a car. With the dimensions given in the figure, calculate the shaded area swept by the wiper. (A) 102.67 cm (B) 205.34 cm (C) 51.33 cm (D) Cannot be determined From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 91 49. Find the area of the quadrilateral ABCD. (Given, \3 = 1.73) (A) 452 sq units (B) 269 sq units (C) 134.5 sq units (D) Cannot sq units 50. The base of a pyramid is a rectangle of sides 18 m x 26 m and its slant height to the shorter side of the base is 24 m. Find its volume. (A) 156\407 (B) 78\407 (C) 312\407 (D) Data insufficient 51. A wire is looped in the form of a circle of radius 28 cm. It is bent again into a square form. What will be the length of the diagonal of the largest square possible thus? (A) 44 cm (B) 44\2 (C) 176/2\2 (D) 88/2\2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 92 SET THEORY Sets: A set is a well defined collection of objects. These are called elements of the set. The sets will be denoted by capital letters A, B, ……… X, Y. The elements of the set are denoted by small letters a, b, ….. x, y. A set can be represented in two ways. (i) Roster form (ii) Set builder form. Example: Let S = {1, 2, 3, 4, 5, 6, 7}. (Roster form) All the elements are written in a curly bracket. S = {x / x e N ; x < 8} (Set builder form) We represent the elements of the set as ‗x‘ and after it we put‖ / ―(such that) and then give the rule which every element of the set should satisfy. If all the elements in set A are in set B then we say A is a subset of B. (A c B). B is super set of A. Number of elements in the set Number of subsets Number of proper subsets 2 2 2 2 2 -2 3 2 3 2 3 -2 4 2 4 2 4 -2 N 2 n 2 n -2 Power set: Let A be any given set. A set which contains all the subsets of A as elements, is called the power set of A. It is denoted by P [A]. Example: A = {p, q, r} P[A] = [{p}, {q}, {r}, {p, q}, {p, r}, {q, r} [p, q, r} |] Finite set and Infinite set: If the number of elements in a set are finite, then it is called a finite set. A set which is not finite is called an infinite set. Equal sets: Two sets are said to be equal if they contain same elements. Ex: A = {1, 2, 3, 4} B = {2, 4, 1, 3} Since the elements in both A and B are equal, they are called equal sets. Equivalent sets: Two sets are to be equivalent if the number of elements in two sets are same. Ex: A = {1, 2, 3}; B = {p, q, r} Number of elements in both the sets are same. So A and B are equivalent sets. If there are not elements in a set, then it is called a null set. It is denoted by { } or |. Union of sets: The set containing the elements of A or B or both is called as union of sets. A B = {x / x e A or x e B} Representing A B by Venn diagrams. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 93 Intersection of sets: The set containing the common elements of set A and set B is called intersection of A and B. A ·B = {x / x e A and x e B} Representing A·B by Venn diagrams: The difference of two sets A – B: The set consisting of all the elements, which belong to A and do not belong to B, is called the difference of A and B. It is denoted by A-B. A-B = {x / x e A and x e B} Similarly B-A = {x / x e B and x e A} A-B = B-A Symmetric difference of two sets: The symmetric difference of two sets is represented by A A B. A A B = (A-B) (B-A) = (AB)-(A·B) Universal set: The union of sets which are to be observed is called universal set and it is denoted by µ. Complementary set: The set of elements which belong to µ and does not belong to A is called complementary of set A , . It is denoted by A , or A c . Basic theorem: If XcY and YcX; then X=Y. In proving equality of two sets we use this basic theorem. This is called Antisymmetric property. (A‘)‘ = A, when A is a subset of some universal st. In the following table A, B, C stand for sets; | the empty set; µ a universal set and p, q, r stand for statements. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 94 ‗÷‘ represents logical equivalence of statements: ‗t‘ to take only the value of ‗T‘ and ‗f‘ to take the only truth value of ‗F‘. S.No. Law Algebra of Sets Statements 1. Idempotent Laws AB = A, A·A=A p v p ÷ p, p ^ p ^ p 2. Associative Laws (AB)C = A (BC) (A·B)·C = A ·(B·C) (p v q) v r ÷ p v (q v r) (p ^ q) ^ r ÷ p ^ (q ^ r) 3. Commutative Laws AB=BA A·B=B·A p v q ÷ q v p p ^ q ÷ q ^ p 4. Distributive Laws A(B·C) = (AB)·(AC) A·(BC) = (A·B)(A·C) p v (q ^ r) ÷ (p v q) ^ (p v r) p ^ (q v r) ÷ (p ^ q) v (p ^ r) 5. Identity Laws AC=A, Aµ=µ A·µ=A, A·C=C p v f ÷ p, p v t ÷ t p ^ f ÷ f, p ^ t ÷ p 6. Complement Laws A A' = µ, A·A' = C (A')' = A, µ' = C, C'=µ p v (~p) ÷ t, p ^ (~P) ÷ f ~ (~p) ÷ p, ~t = f, ~ f ÷ t 7. De Morgan‘s Laws (AB)' = A' · B' (A·B)' = A'B' ~(p v q) ÷ (~p) ^ (~q), ~(p ^ q) ÷ (~p) v (~q) In any law of equality of sets, if we interchange and ·; and µ and |; the resulting law would also be true. This principle is known as Principle of duality. Example: A| = A ¬A·µ = A If A · B = |; then a and B are called disjoint sets. A · B | ¬A c B' and B c A'. A and B are two subsets of a universal set µ. Then A · B = A - B' = B - A'. If A c B then A' B'. [If A is a subset of B, then A' is superset of B'] A' - B' = B – A A B = | ¬ A = | and B = | If A c B ; B c C; then A c C (Transitive property). A – (A – B) = A · B. A B = A · B · A = B. If A c B then A (B – A) = B. If A and B are disjoint sets, then n (A B) = n (A) + n (B). If A and B are any two non-empty sets, then n (A B) = n (A) + n (B) – n (A · B). N (A B C) = n (A) + n (B) + n (C) – n(A · B) – n(B · C) – n(C· A) + n(A · B · C) l and m are two coplanar lines. If l · m = |; then the lines l and m are parallel to each other. EXERCISE From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 95 1. If A B = A · B, then A) A c B B) B c A C) A B D) A = B [ ] 2. {x/x e N; 4 < x < 7} = A) {4, 5, 6, 7} B) {4, 5, 6} C) {5, 6} D) {5, 6, 7} [ ] 3. If A c B; then A – B = A) A B) B C) µ D) | [ ] 4. If A c B; then A · B = A) A B) B C) | D) µ [ ] 5. A | = A) A B) B C) | D) A ( [ ] 6. A A ( A) µ B) A C) A ( D) | [ ] 7. (A B) C = A (B C) this is A) Distributive law B) De – Morgan‘s law C) Associative law D) Commutative law [ ] 8. A · A ( A) µ B) A C) A ( D) | [ ] 9. If A c B; then A B = A) B B) A C) µ D) A · B [ ] 10. If A, B are non- empty sets; then A – B = A) A B) B - A C) A · B ( D) A B ( [ ] 11. If A c B; then A ( B = A) A From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 96 B) B C) A · B D) A B [ ] 12. If A c B and B c A; then A) A = B B) A = B C) A = B = | D) A B [ ] 13. A) {1, 2, 3} B) {4, 6} C) {3, 4, 6} D) {1, 2} [ ] 14. (A – B) (B – A) = A) | B) A C) µ D) A A B [ ] 15. (A – B) · (B – A) = A) | B) A C) B D) A A B [ ] 16. If A _ B; n (A) = 10; n (B) = 15; then n (A · B) = A) 5 B) 25 C) 10 D) 15 [ ] 17. Venn diagram that represents (A – B) is 18. If A = {3, 5, 7, 9}; B = {3, 5, 6}; C = {1, 2, 7} the A – (B C) = A) {9} B) {1, 2} C) {1, 2, 9} D) | [ ] 19. If A = {p, q, r}; B = {x, y, z} then A and B are A) subsets B) null sets From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 97 C) equal D) equivalent sets [ ] 20. If there are 64 elements in the powerset of A, then number of elements in the set A are A) 4 B) 16 C) 32 D) 6 [ ] 21. If there are 4 elements in set A, then the number of elements in P [A] are A) 8 B) 16 C) 32 D) 4 [ ] 22. If l and m are two straight lines and l · m = |; then l and m are A) Perpendicular B) Parallel C) Intersecting lines D) one and the same [ ] 23. A) Parallelogram B) Rhombus C) quadrilateral D) Trapezium [ ] 24. A · (µ - A) = A) µ B) A C) A ( D) | [ ] 25. A (A A ( ) = A) µ B) | C) A D) A ( [ ] 26. A – (B B ( ) = A) µ B) | C) B D) B ( [ ] 27. If A – B = |; then A) A = B B) B = | C) A c B D) B c A [ ] 28. If (A ( ) ( = |; then A = A) µ B) | C) A ( D) | ( [ ] 29. A = {4, 5, 6}; B = {7, 8, 9}; then (A B) ( A) {4, 5, 6, 7, 8, 9} B) | From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 98 C) µ D) cannot be decided since µ is not given E) | ( [ ] 30. x e (A B) ¬ A) x e A v x e B B) x e A . x e B C) x e A; x e B D) x e A; x e B [ ] 31. If µ = {1, 2, 3, 4, 5, 6, 7}; A = {2, 5, 7}; B = {1, 3, 4} then (A B) ( A) {1, 2, 3, 4, 5, 7} B) {6} C) {1, 3, 4, 6, 7} D) | [ ] 32. µ ( = | is A) De-Morgan‘s law B) Idempotent law C) Complement law D) Identity law [ ] 33. If n (A) = 37; n (A B) = 52; n (A · B) = 8 then n (B) = A) 23 B) 29 C) 44 D) 45 [ ] 34. The shaded area represents A) A · (B · C) B) A (B · C) C) A (B C) D) A · (B C) [ ] 35. From the Venn diagram n (M P C) A) 28 B) 18 C) 26 D) 20 [ ] From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 99 PERMUTATIONS AND COMBINATIONS 1. Principle of Counting: If some procedure can be performed in n 1 different ways and if, following procedure, a second procedure can be performed in n 2 different ways, and if, following this second procedure, a third procedure can be performed in n 3 different ways, and so for; then the number of ways the procedures can be performed in the order indicated is the product n 1 , n 2 , n 3 ……………. 2. The number of permutations of n distinct objects taken r (0srsn) at a time is given by n!/(n-r)! ; n > r where n P r = 0 ; n<r 3. The number of permutations of n objects taken all together, when p 1 of the objects are alike of one kind, p 2 of them are alike and of the second kind,…., p r of them are alike and of the rth kind, where p 1 + p 2 + ….. p r = n is given by n!/p 1 ! P 2 !.....p r ! 4. The number of combinations of n distinct objects taken r (0 s r s n) at a time is given by n n!/(n-r)! r!, n > r = C(n, r) = r 0 , n < r 5. Some Result to Remember (i) n C 0 = 1 = n C n (ii) n C r = n C n-r (0 s r s n) (iii) n C r-1 + n C r = n+1 C r (1 s r s n) (iv) n C r = n C s implies r = s or r + s = n. (v) n C r = {n-r+1)/r} n C r -1 r = n/r, n even (vi) n C r is greatest = r = n±1/2, n odd (vii) n C0 + n C 1 + n C2 + ….+ n C n = 2 n . (viii) n C 0 + n C 2 + …. = n C 1 + n C 3 +……=2 n-1 . (ix) 2 n+1 C 0 + 2 n+1 C 1 +….+2 n+1 C n =2 2n . (x) The number of combinations of n distinct objects taken r(s n) at a time, when k(0 s k s r) particular objects always occur, is n-k C r-k . (xi) The number of combinations of n distinct objects taken r at a time, when k(1 s k s n) never occur, is n+k C r . (xii) The total number of selections of one or more objects from n different objects = 2 n – 1 = ( n C 1 + n C 2 + n C 3 +….+ n C n ). (xiii) r C r + r+1 C r +….+ n C r = n+1 C r+1 6. The total number of selection of any number of things from n identical things n + 1 , (when selection of 0 things is allowed) ¬ n , (when at least one thing is to be selected) 7. The total number of selections from p like things, q like things of another type and r distinct things = (p + 1) (q + 1) 2 r – 1 (if at least one thing is to be selected) (p + 1) (q + 1) 2r – 2 (if none or all cannot be selected) From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 100 8. The total number of selections of r tings from n different things when each thing can be repeated unlimited number of times = n+r-1 C r . 9. The number of ways to distribute n different things between two persons, one receiving p things and the other q things, where p + q = n ¬ n C p × n-p C q = n!/p!(n – p)! × (n - p!/q!(n – p – q)! = n!/p!q! {n = p + q} Similarly for 3 persons, the number of ways = n!/p! q! r!, where p + q + r = n. 10. The number of ways to distribute m × n different things among n persons equally = (nm)!/(m!)n. 11. The number of ways to divide n different things into three bundles of p, q and r things = n!/p! q! r! . 1/3!. 12. The number of ways to divide m × n different things into n equal bundles = (mn)!/(m!)n . 1/n! 13. The total number of ways to divide n identical things among r persons = n+r-1 C r-1 . 14. The number of ways in which n different objects can be distributed into r different boxes, blank boxes being admissible is r n . 15. The number of ways in n different objects can be distributed into r different boxes are not allowed, is coefficient of · n in n! (e · -1)r. 16. The number of circular arrangements of n different things (n – 1)! 17. When clockwise and anticlockwise arrangements are not different, number of circular arrangements of n different things = ½(n – 1)! 18. Types of Permutations based upon Geometrical Applications: (i) Out of n non-concurrent and non-parallel straight lines points of intersection are = n C 2 (ii) Out of ‗n‘ points the number of straight lines are (when no three are collinear) = n C 2 (iii) If out of n points m are collinear, then Number of straight lines = n C 2 – m C 2 + 1 (iv) To find number of diagonals Number of diagonals = n(n – 3)/2 (v) Number of triangle formed from n points (when no three points are collinear) (vi) Number of triangles out of n points in which m are collinear = n C 3 – m C 3 (vii) Number of triangles that can be formed out of n points (when none of the side is common to the sides of polygon) = n C 3 – n C 1 – n C 1 . n-4 C 1 (viii) Number of parallelogram in two system of parallel lines (when I set contains m parallel lines and II set contains n parallel lines) = n C 2 × m C 2 (ix) Number of squares m-1 = E (m – r) (n – r) ; (m < n) r=1 EXERCISE From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 101 1. The number of arrangements of letters of the word BANANA in which the two N‘s do not appear adjointly is: (a) 40 (b) 60 (c) 80 (d) 100 2. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that odd digits occupy even positions? (a) 16 (b) 36 (c) 60 (d) 180 3. ***Ten different letters of an alphabet are given. Words with 5 letters are formed from these given letters. Then the number of words which have atleast one letter repeated is: (a) 69760 (b) 30240 (c) 99748 (d) None of these 4. How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, repetition of digits being allowed? (a) 374 (b) 375 (c) 376 (d) None of these 5. The straight lines l 1 , l 2 , l 3 are parallel and lie in the same plane. A total number of m points are taken on l 1 , n points on l 2 , k points on l 3 . the maximum number of triangle formed with vertices of these points are : (a) m+n+k C 3 (b) m+n+k C 3 – m C 3 – n C 3 – k C 3 (c) m C 3 + n C 3 + k C 3 (d) None of these 6. There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is: (a) 10 2 (b) 1023 (c) 2 10 (d) 10! 7. How many 10 digit numbers can be written by using the digits 1 and 2? (a) 10 C 1 + 9 C 2 (b) 2 10 (c) 10 C 2 (d) 10! 8. Number of divisors of the form 4n + 2 (n > 0) of the integer 240 is: (a) 4 (b) 8 (c) 10 (d) 3 9. A five-digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways this can be done is: (a) 216 (b) 240 (c) 600 (d) 3125 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 102 - 10. The number of positive integers which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 by using each digit not more than once in each number is: (a) 1200 (b) 1500 (c) 1600 (d) 1630 11. The total number of 9 digit numbers which have all different digits is: (a) 10! (b) 9! (c) 9.9! (d) 10.10! 12. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4, and then the men select the chairs from amongst the remaining. The number of possible arrangements is: (a) 4 C 3 × 4 C 2 (b) 4 C 2 × 4 P 3 (c) 4 P 2 × 4 P 3 (d) None of these 13. From 4 officers and 8 Jawans, a committee of 6 is to be chosen to include exactly one officer. The number of such committees is: (a) 160 (b) 200 (c) 224 (d) 300 14. Given 5 different green dyes, 4 different blue dyes and 3 different red dyes. The number of combinations of dyes which can be chosen taking at least one green and one blue dye is: (a) 3600 (b) 3720 (c) 3800 (d) None of these 15. If a polygon has 44 diagonals, then the number of its sides are : (a) 11 (b) 7 (c) 8 (d) None of these 16. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is: (a) 6 (b) 7 (c) 8 (d) 9 17. The number of five-digit telephone numbers having at least one of their digits repeated is: (a) 90000 (b) 100000 (c) 30240 (d) 69760 18. If n is an integer between 0 and 21, then the minimum value of n! (21-n)! is: (a) 9! 21! (b) 10! 11! (c) 20! (d) 21! 19. The total number of six digit numbers x 1 x 2 x 3 x 4 x 5 x 6 having the property x 1 <x 2 sx 3 <x 4 <x 5 sx 6 , is equal to: From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 103 - (a) 10 C 6 (b) 12 C 6 (c) 11 C 6 (d) None of these 20. The total number of three digit numbers, then sum of whose digits is even, is equal to: (a) 450 (b) 350 (c) 250 (d) 325 21. If letters of the work ‗KUBER‘ are written in all possible orders ad arranged as in a dictionary, then rank of the word ‗KUBER‘ will be: (a) 67 (b) 68 (c) 65 (d) 69 22. In a chess tournament, all participants were to play one game with the another. Two players fell ill after having played 3 games each. If total number of games played in the tournament is equal to 84, then total number of participants in the beginning was equal to: (a) 10 (b) 15 (c) 12 (d) 14 23. In a country no two persons have identical set of teeth and there is no person with out a tooth, also no person has more than 32 teeth. If shape and size of tooth is disregarded and only the position of tooth is considered, then maximum population of that country can be: (a) 2 32 (b) 2 32 -1 (c) can‘t be obtained (d) none of these 24. The total number of flags with three horizontal strips, in order, that can be formed using 2 identical red, 2 identical green and 2 identical white strips, is equal to: (a) 4! (b) 3.(4!) (c) 2.(4!) (d) None of these 25. The sides AB, BC, CA of a triangle ABC have 3, 4, 5 interior points respectively on them. Total number of triangles that can be formed using these points as vertices, is equal to: (a) 135 (b) 145 (c) 178 (d) 205 26. ‗n‘ different toys have to be distributed among ‗n‘ children. Total number of ways in which these toys can be distributed so that exactly one child gets no toy, is equal to: (a) n! (b) n! n C 2 (c) (n-1)! n C 2 (d) n! n-1 C 2 27. Total number of non-negative integral solutions of x 1 +x 2 +x 3 = 10 is equal to: (a) 12 C 3 (b) 10 C 3 (c) 12 C 2 (d) 10 C 2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 104 - 28. A class contains 3 girls and four boys. Every Saturday five student go on a picnic, a different group of students is being sent each week. During the picnic, each girl in the group is given doll by the accompanying teacher. All possible groups of five have gone once, the total number of dolls the girls have got is: (a) 21 (b) 45 (c) 27 (d) 24 29. total number of 4 digit number that are greater than 3000, that can be formed using the digits 1, 2, 3, 4, 5, 6 (no digit is being repeated in any number) is equal to: (a) 120 (b) 240 (c) 480 (d) 80 30. A variable name in certain computer language must be either a alphabet or a alphabet followed by a decimal digit. Total number of different variable names that can exist in that language is equal to: (a) 280 (b) 290 (c) 286 (d) 296 31. There are 10 person among whom two are brother. The total number of ways in which these persons can be seated around a round table so that exactly one person sit between the brothers, is equal to: (a) (2!) (7!) (b) (2!) (8!) (c) (3!) (7!) (d) (3!) (8!) From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 105 - PROBABILITY Experiment An operation which results in some well-defined outcomes is called an experiment. Random Experiment An experiment whose outcome cannot be predicted with certainty is called a random experiment. In other words, if an experiment is performed many times under similar conditions and the outcome of each time is not the same, then this experiment is called a random experiment. Example: a) Tossing of a fair coin b) Throwing of an unbiased die c) Drawing of a card from a well shuffled pack of 52 playing cards Sample Space The set of all possible outcomes of a random experiments is called the sample space for that experiment. It is usually denoted by S. Example: f) When a die is thrown, any one of the numbers 1, 2, 3, 4, 5, 6 can come up. Therefore. Sample space S = {1, 2, 3, 4, 5, 6} g) When a coin is tossed either a head or tail will come up, then the sample space w.r.t. the tossing of the coin is S = {H, T} h) When two coins are tossed, then the sample space is Sample point / event point Each element of the sample spaces is called a sample point or an event point. Example: When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6} where 1, 2, 3, 4, 5 and 6 are the sample points. Discrete Sample Space A sample space S is called a discrete sample if S is a finite set. Event A subset of the sample space is called an event. Problem of Events  Sample space S plays the same role as universal set for all problems related to the particular experiment.  | is also the subset of S and is an impossible Event.  S is also a subset of S which is called a sure event or a certain event. Types of Events A. Simple Event/Elementary Event An event is called a simple Event if it is a singleton subset of the sample space S. Example: a) When a coin is tossed, then the sample space is S = {H, T} Then A = {H} occurrence of head and B = {T} occurrence of tail are called Simple events. b) When two coins are tossed, then the sample space is S = {(H,H); (H,T); (T,H); (T,T)} Then A = {(H,T)} is the occurrence of head on 1 st and tail on 2 nd is called a Simple event. B. Mixed Event or Compound Event or Composite Event From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 106 - A subset of the sample space S which contains more than one element is called a mixed event or when two or more events occur together, their joint occurrence is called a Compound Event. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 107 - Example: When a dice is thrown, then the sample space is S = {1, 2, 3, 4, 5, 6} Then let A = {2, 4 6} is the event of occurrence of even and B = {1, 2, 4} is the event of occurrence of exponent of 2 are Mixed events Compound events are of two type: a) Independent Events, and b) Dependent Events C. Equally likely events Outcomes are said to be equally likely when we have no reason to believe that one is more likely to occur than the other Example: When an unbiased die is thrown all the six faces 1, 2, 3, 4, 5, 6 are equally likely to come up. D. Exhaustive Events A set of events is said to be exhaustive if one of them must necessarily happen every time the experiments is performed. Example: When a die is thrown events 1, 2, 3, 4, 5, 6 form an exhaustive set of events. Important We can say that the total number of elementary events of a random experiment is called the exhaustive number of cases. E. Mutually Exclusive Events Two or more events are said to be mutually exclusive if one of them occurs, others cannot occur. Thus if two or more events are said to be mutually exclusive, if not two of them can occur together. Hence, A 1 , A 2 , A 3 ,…, An are mutually exclusive if and only if A i ·A j = | ¬ i = j Example: a) When a coin is tossed the event of occurrence of a head and the event of occurrence of a tail are mutually exclusive events because we cannot have both head and tail at the same time. b) When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6} Let A is an event of occurrence of number greater than 4 i.e., {5, 6} B is an event of occurrence of an odd number {1, 3, 5} C is an event of occurrence of an even number {2, 4, 6} Here, events B and C are Mutually Exclusive but the event A and B or A and C are not Mutually Exclusive. F. Independent Events or Mutually Independent events Two or more event are said to be independent if occurrence or non-occurrence of any of them does not affect the probability of occurrence of or non-occurrence of their events. Thus, two or more events are said to be independent if occurrence or non-occurrence of any of them does not influence the occurrence or non-occurrence of the other events. Example: Let bag contains 3 Red and 2 Black balls. Two balls are drawn one by one with replacement. Let A is the event of occurrence of a red ball in first draw. B is the event of occurrence of a black ball in second draw. then probability of occurrence of B has not been affected if A occurs before B. As the ball has been replaced in the bag and once again we have to select one ball out of 5(3R + 2B) given balls for event B. G. Dependent Events Two or more events are said to be dependent, if occurrence or non-occurrence of any one of them affects the probability of occurrence or non-occurrence of others. Example: Let a bag contains 3 Red and 2 Black balls. Two balls are drawn one by one without replacement. Let A is the event of occurrence of a red ball in first draw B is the event of occurrence of a black ball in second draw. In this case, the probability of occurrence of event B will be affected. Because after the occurrence of event A i.e. drawing red ball out of 5(3R + 2B), the ball is not replaced in bag. Now, for the event B, we will have to draw 1 black ball from the remaining 4(2R + 2B) balls which gets affected due to the occurrence of event A. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 108 - H. Complementary Events Let S be the sample space for a random experiment and let E be the event. Also, Complement of event E is denoted by E‘ or E, where E‘ means non occurrence of event E. Thus E‘ occurs if and only if E does not occur. n(E) + n(E‘) = n(S) Occurrence of an Event For a random experiment, let E be an event Let E = {a, b, c}. If the outcome of the experiment is either a or b or c then we say the event has occurred. Sample Space : The outcomes of any type Event : The outcomes of particular type Probability of Occurrence of an event Let S be the same space, then the probability of occurrence of an event E is denoted by p(E) and is defined as P(E) = n(E)/n(S) = number of elements in E/number of elements in S P(E) = number of favourable/particular cases total number of cases Example: a) When a coin is tossed, then the sample space is S = {H, T} Let E is the event of occurrence of a head ¬ E = {H} b) When a die is tossed, sample space S = {1, 2, 3, 4, 5, 6} Let A is an event of occurrence of an odd number And B is an event of occurrence of a number greater than 4 ¬ A = {1, 3, 5} and B = {5, 6} P(A) = Probability of occurrence of an odd number = n(A)/n(S) = 3/6 = ½ and P(B) = Probability of occurrence of a number greater than 4 = n(B)/n(S) = 2/6 = 1/3 EXERCISE Questions 1 – 7 A coin is flipped three. Find the probability of getting 1. A head exactly once. 1. 1/8 2. 1/ 4 3. 3/8 4. 1/ 2 2. tails exactly twice. 1. 1 /4 2. 3/8 3. 1/8 4. None of these 3. heads all three times. 1. 1/8 2. ¼ 3. 3/8 4. 1/ 2 4. a tail at least once. 1. 1 /4 2. 7/8 3. 1/8 4. 3/8 5. a head at least two times. S 1. 3/8 2. 1 /4 3. 1/8 4. 1 /2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 109 - 6. a head in the first throw, a tail in the second, and a head in the third. 1. 1 /8 2. 1/ 4 3. 3/8 4. 1 /2 7. a head in the third toss, if in the first two tosses the coin landed tails. 1. 1/ 2 2. 1/ 8 3. 7/8 4. 1 /4 Questions 8 – 11 From a pack of 52 cards, a card is chosen at random. Find the probability of it being 8. a knave. 1. 1/13 2. 1 /4 3. 1 /2 4. 1/52 9. a card of hearts. 1. 1 /2 2. 1 /4 3. 1 /2 4. 1/52 10. a 7 of clubs. 1. 1/26 2. 5/52 3. 1/13 4. 1/52 11. a king of diamonds or hearts. 1. 1/13 2. 3/26 3. 1/26 4. 1 /4 Questions 12 – 16 A bag contains 5 red, 6 blue and 9 black balls. Find the probability that a ball drawn at random 12. is either red or blue or black. 1. 0 2. 19/20 3. 21/20 4. 20/20 13. is blue. 1. 1 /4 2. 7/10 3. 3/10 4. 9/20 14. is red or blue. 1. 11/20 2. 1 /4 3. 3 /4 4. 7/10 15. is blue or black. 1. 11/20 2. 1 /4 3. 3 /4 4. 7/10 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 110 - 16. is not blue. 1. 1 /4 2. 7/10 3. 3/10 4. 9/20 17. Two cards are drawn successively with replacement from a well shuffled deck of 52 cards, the probability that both of these will be aces is: 1. 1/169 2. 1/201 3. 1/2652 4. 4/663 18. Find the probability that there are 53 Mondays in Leap year. 1. 1/7 2. 2/7 3. 3/7 4. 4/7 19. if P (A)=0.3 P(B)=0.4 and P(AB)=0.6, then find P(A·B). 1. 0.3 2. 0.4 3. 0.1 4. 0.5 20. In throwing a fair dice, what is the probability of getting the number ‗3‘? 1. 1/3 2. 1/6 3. 1/9 4. 1/12 21. What is the number of throwing a number greater than 4 with an ordinary dice whose faces are numbers from 1 to 6. 1. 1 /3 2. 1/6 3. 1/9 4. 1/12 22. Three coins are tossed. What is the probability of getting (i) 2 Tails and 1 Head 1. 1 /4 2. 3/8 3. 2/3 4. 1/ 8 (ii) 1 Tail and 2 Heads 1. 3 /8 2. 1 3. 2 /3 4. 3 /4 23. Three coins are tossed. What is the probability of getting (i) neither 3 Heads nor 3 Tails 1. 1 /2 2. 1 /3 3. 2/3 4. 3 /4 (ii) three heads 1. 1 /8 2. 1 /4 3. 1 /2 4. 2/ 3 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 111 - 24. What is the probability of throwing a number greater than 2 with a fair dice 1. 2 /3 2. 2/5 3. 1 4. 3/5 25. A can hit the target 3 times in 6 shots, B 2 times in 6 shots and C 4 times in 6 shots. They fire a volley. What is the probability that at least 2 shots hit? 1. 1 /2 2. 1/3 3. 2 /3 4. 1/ 4 26. If 4 whole numbers are taken at random and multiplied together, what is the chance that the last digit in the product is 1, 3, 7 or 9 ? 1. 15/653 2. 12/542 3. 16/625 4. 13/625 27. A life insurance company insured 25,000 young boys, 14,000young girls and 16,000young adults. The probability of death within 10 years of a young boy, young girl and a young adult arte 0.02, 0.03 and 0.15 respectively. One of the insured persons dices. What is the probability that the dead person is a young boy? 1. 36/165 2. 25/166 3. 26/165 4. 30/165 28. A team of 4 is to be constituted out of 5 girls and 6 boys. Find the probability that the team may have 3 girls. 1. 4 /11 2. 3/11 3. 5/11 4. 2/11 29. 12 persons are seated around a round table. What is the probability that two particular persons sit together? 1. 2/11 2. 1/6 3. 3/11 4. 3/15 30. Six boys and six girls sit in a row randomly. Find the probability that all the six girls sit together. S 1. 3/22 2. 1/132 3. 1/1584 4. 1/66 31. A bag contains 5 red, 4 green and 3 black balls. If three balls are drawn out of it at random, find the probability of drawing exactly 2 red balls. 1. 7/22 2. 10/33 3. 7/12 4. 7/11 32. A bag contains 100 tickets numbered 1, 2, 3,…., 100. If a ticket is drawn out of it at random, what is the probability that the ticket drawn has the digit 2 appearing on it. 1. 19/100 2. 21/100 3. 32/100 4. 23/100 33. A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals: 1. 1 /2 2. 1/32 3. 31/32 4. 1/5 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 112 - 34. Three mangoes and three apples are kept in a box. If two fruits are selected at random from the box, the probability that the selection will contain one mango and one apple, is: 1. 3/5 2. 5/6 3. 1/36 4. None of these From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 113 - PROGRESSIONS 1. Sequence Sequence is a function whose domain is the set N of natural numbers. Real Sequence: A sequence whose range is a subset of R is called a real sequence. Series: If a 1 , a 2 , a 3 , a 4 ,……, a n ,….. is a sequence, then the expression a 1 + a 2 + a 3 + a 4 + a 5 + ….+ a n + ….is a series. A series is finite or infinite according as the number of terms in the corresponding sequence is finite or infinite. Progressions: It is not necessary that the terms of a sequence always follow a certain patterns are called progressions. 2. Arithmetic Progression (A.P.) A sequence is called an arithmetic progression if the difference of a term and the previous term is always same, i.e. a n+1 – a n = constant (=d) for all n e N The constant difference, generally denoted by d is called the common difference. For example. Show that the sequence <a n > is an A.P. if its nth term a linear expression in n and in such a case the common difference is equal to the coefficient of n. Solution. Let <a n > be a sequence such that its nth term is a linear expression in n i.e. a n = An + B where A, B are constants. a n+1 = A(n + 1) + B a n+1 – an = {A (n + 1) + B} – {An + B} = A, i.e. coefficient of n. 3. Properties of an Arithmetic Progression (i) If a is the first term and d the common difference of an A.P., then its nth term a n is given by a n = a + (n + 1) d (ii) A sequence is an A.P. iff its nth term is of the form A n + B i.e. a linear expression in n. the common difference in such a case is A i.e. the coefficient of n. (iii) If a constant is added to or subtracted from each term of an A.P., then the resulting sequence is also an A.P. with the same common difference. (iv) If each term of given A.P. is multiplied or divided by a non-zero constant k, then the resulting sequence is also an A.P. with common difference kd or d/k, where d is the common difference of the given A.P. (v) In a finite A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term i.e. (vi) Three numbers a, b, c are in A.P. iff b = a + c. (vii) If the terms of an A.P. are chosen at regular intervals, then they form an A.P. (viii) If a n , a n+1 and a n+2 are three consecutive terms of an A.P., then 2a n+1 = a n +a n+2 4. Selection of Terms in an A.P. It should be noted that in case of an odd number of terms, the middle term is a and the common difference is d while in case of an even number of terms the middle terms are a – d, a + d and the common differences is 2d. i.e. (i) Selecting two terms of A.P. are a – d, a + d. (ii) Selecting four terms of A.P.: a – 3d, a – d, a + d, a + 3d and so on….. (iii) Selecting 3 terms of A.P.: a – d, a, a + d (iv) Selecting 5 terms of A.P.: a – 2d, a – d, a, a + d, a + 2d. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 114 - 5. Some Useful Results 6. Sum to n Terms of an A.P. The sum S n of n terms of an A.P. with first term ‗a‘ and common difference ‗d‘ is given by S n = (n/2)[2a + (n – 1)d] Also, S n = (n/2)[a + l], where l = last term = a + (n – 1)d. Note. A sequence is an A.P. if and only if the sum of its n terms is of the form A n 2 + Bn, where A, B are constants. In such a case, the common difference of the A.P. is 2A. 7. Insertion of Arithmetic Means If between two given quantities a and b we have to insert n quantities A 1 , A 2 ,…, A n such that a, A 1 , A 2 ,…A n , b form an A.P., then we say that A 1 , A 2 ,…, A n are arithmetic means between a and b. Insertion of n Arithmetic Means between a and b Let A 1 , A 2 ,…,A n be n arithmetic means between two quantities a and b. Then, a, A 1 , A 2 ,…, A n , b is an A.P. Let d be the common difference of this A.P. Clearly, it contains (n + 2) terms. These are the required arithmetic means between a and b. 8. Geometric Progression A sequence of non-zero numbers is called a geometric progression (or G.P.) if the ratio of a term and the term proceeding to it is always a constant quantity. The constant ratio is called the common ratio of the G.P. In other words, a sequence a 1 , a 2 , a 3 ,…, a n ,…is called a geometric progression if (a n+1 ) / a n = constant for all n e N For example. Show that the sequence given by a n = 3 (2 n ). for all n e N, is a G.P. Also, find its common ratio. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 115 - Solution. We have, a n = 3 (2 n ) a n +1 = 3 (2 n +1) Now, (a n+1 )/an = 3(2 n+1 )/3(2 n ) =2 Clearly, (a n+1 )/an = 2 (constant), for all n e N. So, the given sequence is an G.P. with common ratio 2. 9. Selection of Terms in G.P. Sometimes if is required to select a finite number of terms in G.P. It is always convenient if we select the terms in the following manner: No. of terms Terms Common ratio 3 a/r, a, ar R 4 a/r 3 , a/r, ar, ar 3 r 2 5 a/r 2 , a/r, a, ar, ar 2 R If the product of the numbers is not given, then the numbers are taken as a, ar, ar 2 , ar 3 ,….. 10. Properties of Geometric Progressions (i) If all the terms of a G.P. be multiplied or divided by the same non-zero constant, then it remains a G.P. with the same common ratio. (ii) The reciprocals of the terms of a given G.P. form a G.P. (iii) If each term of a G.P. be raised to the same power, the resulting sequence also forms a G.P. (iv) In a finite G.P. the product of the terms equidistant form the beginning and the end is always same and is equal to the product of the first and the last term. (v) Three non-zero numbers, a, b, c are in G.P. iff b 2 = ac. (vi) If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms a G.P. (vii) If a 1 , a 2 , a 3 ,…, a n ,…is a G.P. of non-zero non-negative terms, then log a 1 , log a 2 ,…, log a n ,… is an A.P. and vice-versa. For example. The third term of a G.P. is 4. Find the product of its first give terms. Solution. Let a be the first term and r the common ratio. Then, a 3 = 4 ¬ ar 2 = 4 ………… (i) Now, Product of first five terms = a 1 a 2 a 3 a 4 a 5 11. Sum of Terms of a G.P. (i) The sum of n terms of a G.P. with first term ‗a‘ and common ratio ‗r‘ is given by (ii) If l is the term of the G.P., then l = ar n-1 (iii) If | r | < 1, then lim r n = 0. Therefore, the sum S of an infinite G.P. with common ratio r satisfying | r | < 1 is given by Thus, the sum S of an infinite G.P. with first term a and common ratio r(-1 < r < 1) is given by From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 116 - EXERCISE 1. How many terms are there in the AP 20, 25, 30,… 130. (a) 22 (b) 23 (c) 21 (d) 24 2. Bobby was appointed to AMS Careers in the pay scale of Rs. 7000 – 500 – 12,500. find how many years he will take top reach the maximum of the scale. (a) 11 years (b) 10 years (c) 9 years (d) 8 years 3. Find the 1 st term of an AP whose 8 th and 12 th terms are respectively 39 and 59. (a) 5 (b) 6 (c) 4 (d) 3 4. A number of squares are described whose perimeters are in GP. Then their sides will be in (a) AP (b) GP (c) HP (d) Nothing can be said 5. There is an AP 1, 3, 5… Which term of this AP is 55? (a) 27 th (b) 26 th (c) 25 th (d) 28 th 6. How many terms are identical in the two APs 1. 3, 5, … up to 120 terms and 3, 6, 9,.. up to 80 terms? (a) 38 (b) 39 (c) 40 (d) 41 7. Find the lowest number in an AP such that the sum of all the terms is 105 and greatest term is 6 times the least. (a) 5 (b) 10 (c) 15 (d) 20 8. Find the 15 th term of the sequence 20, 15, 10, … (a) -45 (b) -55 (c) -50 (d) 0 9. A number 15 divided into three parts which are in AP and the sum of their squares is 83. Find the smallest number. (a) 5 (b) 3 (c) 6 (d) 8 10. The sum of the first 16 terms of an AP whose first term and third term are 5 and 15 respectively is (a) 600 (b) 765 (c) 640 (d) 680 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 117 - 11. The number of terms of series 54 + 51 + 48 + .. such that the sum is 513 is (a) 18 (b) 19 (c) Both a and b (d) None of these 12. The least value of n for which the sum of the series 5 + 8+ 11…n terms is not less than 670 is (a) 20 (b) 19 (c) 22 (d) 21 13. A man receives Rs. 60 for the first week and Rs. 3 more each week than the preceding week. How much does he earn by the 20 th week? (a) Rs. 1770 (b) Rs. 1620 (c) Rs. 1890 (d) None of these 14. How many terms are there in the GP 5, 20, 80, …20, 480? (a) 6 (b) 5 (c) 7 (d) 8 15. A boy agrees to work at the rate of one rupee on the first day, two rupees on the second day, four rupees on the third day, and so on. How much will the boy get if he starts worki8ng on the 1 st February and finishes on the 20 th of February? (a) 2 20 (b) 2 20 – 1 (c) 2 19 – 1 (d) 2 19 16. If the fifth term of a GP is 81 and first term is 16, what will be the 4 th term of GP? (a) 36 (b) 18 (c) 54 (d) 24 17. The seventh term of a GP is 8 times the fourth term. What will be the first term when it‘s fifth term is 48? (a) 4 (b) 3 (c) 5 (d) 2 18. The sum of three numbers in a GP is 14 and the sum of their squares is 84. Find the largest number. (a) 8 (b) 6 (c) 4 (d) None of these 19. The first term of an arithmetic progression is unity and the common difference is 4. which of the following will be a term of this AP? (a) 4551 (b) 10091 (c) 7881 (d) 13531 20. How many natural numbers between 300 to 500 are multiple of 7? (a) 29 (b) 28 (c) 27 (d) None of these From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 118 - 21. The sum of the first and the third term of a geometric progression is 20 and the sum of its first three terms is 26. Find the progression. (a) 2, 6, 18,.. (b) 18, 6, 2,.. (c) Both of these (d) Cannot be determined 22. If a man saves Rs. 4 more each year than he did the year before and if he saves Rs. 20 in the first year, after how many years will his savings be more than Rs. 1000 altogether? (a) 19 years (b) 20 years (c) 21 years (d) 18 years 23. A man‘s salary is Rs. 800 per month in the first year. He has joined in the scale of 800 – 40 – 1600. After how many years will his savings be Rs. 64,800? (a) 8 years (b) 7 years (c) 6 years (d) Cannot be determined 24. The 4 th and 10 th term of an GP are 1/3 and 243 respectively. Find the 2 nd term. (a) 3 (b) 1 (c) 1/27 (d) 1/9 25. The 7 th and 21 st terms of an AP are 6 and -22 respectively. Find the 26 th term. (a) -34 (b) -32 (c) -12 (d) -10 26. The sum of 5 numbers in AP is 30 and the sum of their squares is 220. Which of the following is the third term? (a) 5 (b) 6 (c) 8 (d) 9 27. Find the sum of all numbers in between 10 – 50 excluding all those numbers which are divisible by 8. (a) 1070 (b) 1220 (c) 1320 (d) 1160 28. The sum of the first four terms of an AP is 28 and sum of the first eight terms of the same AP is 88. Find the sum of the first 16 terms of the AP? (a) 346 (b) 340 (c) 304 (d) 268 29. Find the general term of the GP with the third term 1 and the seventh term 8. (a) (2 3/4 ) n-3 (b) (2 3/2 ) n-3 (c) (2 3/4 ) 3-n (d) None of these 30. Find the number of terms of the series 1/81, -1/27, 1/9,… -729. (a) 11 (b) 12 (c) 10 (d) 13 31. Four geometric means are inserted between 1/8 and 128. find the third geometric mean. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 119 - (a) 4 (b) 16 (c) 32 (d) 8 32. A and B are two numbers whose AM is 25 and GM is 7. Which fo the following may be a value of A? (a) 10 (b) 20 (c) 49 (d) 25 33. Two numbers A and B are such their GM is 20% lower than their Am. Find the ratio between the numbers. (a) 3 : 2 (b) 4 : 1 (c) 2 : 1 (d) 3 : 1 34. A man saves Rs. 100 in January 2002 and increases his savings by Rs. 50 every month over the previous month. What is the annual savings for the man in the year 2002? (b) Rs. 4200 (c) Rs. 4500 (d) 4000 (e) None of these From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 120 - STATISTICS 1. Arithmetic Mean (i) Mean of unclassified data. Let x 1 , x 2 , …., x n be n observations, then their arithmetic mean is given by x = x1+x2+…..+xn/n = 1/n ¿ = n i xi 1 (ii) Mean of grouped data. Let x 1 , x 2 , x 3 ,…, x n be n observations and let f 1 , f 2 ,…, f n be their corresponding frequencies, then their arithmetic mean is given by ¿ ¿ = = = + + + + + + = n i i n i i i n n n f x f f f f x f x f x f x 1 1 2 1 2 2 1 1 _ .... ... 2. Weighted Arithmetic Mean If w 1 , w 2 , w 3 ,…, w n are the weights assigned to the values x 1 , x 2 , x 3 , ,…, x n respectively, then the weighted average is defined as: n n n w w w x w x w x w M WeightedA + + + + + = .... .... . . 2 1 2 2 1 1 3. Combined Mean If we are given the A.M. of two data sets and their sizes, then the combined A.M. of two data sets can be obtained by the formula 2 1 2 _ 2 1 _ 1 12 _ n n x n x n x + + = where x 12 = Combined mean of the two data sets 1 and 2 x 1 = mean of the first data x 2 = Mean of the second data n 1 = Size of the first data n 2 = Size of the second data. 4. Geometric Mean If x1, x2, x3 are n observations, none of them being zero, then their geometric mean is defined as G.M. = (x 1 , x 2 , x 3 ….x n ) 1/n 5. Harmonic Mean The harmonic mean of n observation x 1 , x 2 ,….., x n is defined as: ¿ ¿ = = | | . | \ | = n i i i n i i x f f M H 1 1 . . 6. Relation among A.M, G.M. And H.M. The arithmetic mean (A.M.), geometric mean (G.M.) and harmonic mean (H.M.) for a given set of observations are related as under: A.M. > G.M. > H.M. Equality sign hold only when all the observations are equal. 7. Median (i) Median of an individual series. Let n be the number of observations. (A) Arrange the data in ascending or descending order. (B) (a) If n is odd, then From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 121 - Median = value of the ½ (n+1)th observation (b) if n is even, then Median = mean of the (n/2)th and (n/2 + 1)th observation (ii) Median of a discrete series (A) Arrange the values of the variate in ascending or descending order. (B) Prepare a commulative frequency table. (C) (a) If n is odd, then Median = size of the ((n+1)/2)th term (b) If n is even, then Median = size of the | | | | . | \ | | . | \ | + + | . | \ | 2 1 2 2 n n th term (iii) median of a Continuous Series (iv) Prepare the commulative frequency table. (B) Find the median class, i.e. the class in which the (n/2)th observation lies. (C) The median value is given by the formula Median = l + , 2 h f c n f × | | | | . | \ | ÷ | . | \ | where l = lower limit of the median class n = total frequency f = frequency of the median class h = width of the median class c f = cumulative frequency of the class preceding the median class. 8. Quartiles, Deciles and percentiles (a) For ungrouped data Quartiles are also a kind of positional averages which divide the complete frequency distribution into four equal parts. Qr = nr/4 ; r = 1, 2, 3 Decline divide the frequency distribution into 10 equal parts. D r = nr/10 ; r = 1, 2, 3…… 9 Percentiles divide it into 100 equal parts P r = nr/100 ; r = 1, 2, 3,…….99 (b) For grouped data Arrange data is ascending order and prepare cumulative frequency 3 , 2 , 1 ; 4 = | . | \ | ÷ + = r f i F Nr l Q r 9 ..... 3 , 2 , 1 ; 10 = | . | \ | ÷ + = r f i F Nr l D r 99 ..... 3 , 2 , 1 ; 100 = | . | \ | ÷ + = r f i F Nr l P r where l is lower limit of the required class, i is class interval, f is frequency of the class and F is sum of all frequencies just above the class of quartile/decile/percentile. 9. Mode From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 122 - (i) Mode of individual series In the case of individual series, the value which is repeated maximum number of times is the mode of the series. (ii) Mode of discrete series In the case of discrete frequency distribution, mode is the value of the variate corresponding to the maximum frequency. (iii) Mode of continuous series (A) Find the modal class, i.e. the class which has maximum frequency. The modal class can be determined either by inspection or with the help of grouping table. (B) The mode is given by the formula Mode = l + (f m – f m -1/2f m – f m -1 – f m+1 ) × h, where l = the lower limit of the modal class h = the width of the modal class f m-1 = the frequency of the class preceding modal class f m = the frequency of the modal class f m+1 = the frequency of the class succeeding modal class. In case, the modal value lies in a class other than the one containing maximum frequency, we take the help of the following formula Mode = l + f m+1 /f m-1 +f m+1 × h, where symbols have usual meaning. A distribution in which mean, median and mode coincide is called a symmetrical distribution. If the distribution is moderately skewed, then mode can be calculated as follows: Mode = 3 Median – 2 Mean. 10. Measures of Dispersion (i) Range It is the difference between the greatest and the smallest observation of the distribution. If L is the largest and S is the smallest observation in a distribution, then its Range = L – S. Also. Coefficient of range = L-S/L+S (ii) Quartile Deviation Quartile deviation or semi-interquartile range is given by Q.D. = ½ (Q 3 – Q 1 ) Coefficient of Q.D. = Q 3 – Q 1 /Q 3 +Q 1 (iii) Mean deviation For a frequency distribution, the mean deviation from an average (median, or arithmetic mean) is given by ¿ ¿ = = ÷ = n i i n i i f x xi f D M 1 1 _ | | . . Coefficient of M.D. = Mean deviation/Corresponding average (iv) Standard deviation The standard deviation of a statistical data is defined as the positive square root of the squared deviations of observations from the A.M. of the series under consideration. (A) Standard deviation (also denoted by o) for ungrouped set of observations is given by N x x f D S n i i i ¿ = ÷ = 1 2 _ ) ( . . o (B) Standard deviation for frequency distribution is given by, N x x f D S n i i i ¿ = ÷ = 1 2 _ ) ( . . From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 123 - where, f i is the frequency of x i (1 s i s n). 11. Skewness We study skewness to have an idea about the shape of the curve which we can draw with the help of the given data. The term ‗skewness‘ refers to lack of symmetry. We can define skewness of a distribution as the tendency of a distribution to depart from symmetry. (i) In a symmetrical distribution, we have mean = median = mode. (ii) When the distribution is not symmetrical, it is called asymmetrical or skewed. In a skewed distribution Mean = Median = Mode. Note: (a) Absolute measures of skewness. (i) S k = mean – median (ii) S k = mean – mode (iii) S k = Q 3 + Q 1 – 2Q 2 or S k = Q 3 + Q 1 – 2 (median). (b) Relative measures of skewness. The following are four important relative measures of skewness: (i) Karl Pearson‘s coefficient of skewness S k = mean – mode/Standard deviation If mode is well defined then using the relation, Mode = 3 median – 2 mean, For a moderately skewed distribution, we get S k = 3 (mean – median)/Standard deviation. If follows that S k = 0, if mean = mode = median. (ii) Empirical relationships. If the data is moderately non-symmetrical, then the following empirical relationships hold: Mean deviation = 4/5 o Semi-inter-quartile range = 2/3 o. Probable error of standard deviation = 2/3 o = Semi-inter-quartile range Quartile deviation = 5/6 M.D. From these relationship, we have 4 S.D. = 5 M.D. = 6 Q.D. EXERCISE 1. From the following frequency table: Marks 0 – 20 20 - 40 40 – 60 60 – 80 80 – 100 Frequency 2 5 13 12 3 (i) Find the number of students securing marks < 60. (ii) Find the number of students securing marks > 80. 1. 20, 3 2. 20, 15 3. 15, 8 4. None of these 2. Find the arithmetic mean of 8, -2, 9, 6, 17, 13, 12. 1. 11 2. 19 3. 9 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 124 - 4. 0 3. Find the arithmetic mean of 5/4, -7/6, 2/3, -4 and 6. 1. 11/20 2. 9/20 3. 13/20 4. 11/10 4. If the arithmetic mean of 7, 6, 5, 9, x, 8, 7 is 6 then find x. 1. 1 2. – 1 3. 0 4. – 2 5. In a distribution, 4, 3, 2, 1 occur with frequencies 1, 2, 3, 4 respectively. Then find the arithmetic mean. 1. 5 2. 4 3. 2 4. 8 6. In a distribution, 6, 4, 8, 3 occur with frequencies 4, 2, 5, 1 respectively. Then find the arithmetic mean. 1. 5.77 2. 6.25 3. 4.5 4. 6.15 7. For the following data, find the arithmetic mean. X 1 2 4 6 12 Frequency 12 6 3 2 1 1. 5 2. 2.5 3. 6 4. 1.5 8. Calculate the mean age (in years). Age (in years) 46 41 45 44 43 43 Number of Persons 5 5 7 12 13 8 1. 44.17 2. 43.46 3. 42.78 4. 41.23 9. The median of the data set 15, 25, 43, 24, 21, 17, 16, and 20 is 1. 21 2. 20.5 3. 20 4. 41 10. Find the median of – 3, - 5, - 8, 0, - 10, 2, 3. 1. 2.5 2. – 2. 5 3. – 3 4. None of these 11. If the median of x/4, x, x/4, x/2, and X/3, (x>0) is 8, then find x. 1. 24 2. 96/7 3. 12 4. None of these 12. Mode of the data set 7, 5, 12, 9, 11, 12, 6, 5, 8, 11, 12, 11, 18, and 11 is 1. 5 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 125 - 2. 11 3. 12 4. 10 13. Find the mode of 10, 9, 8, 3, 10, 11, 10. 1. 14 2. 8 3. 12 4. 10 14. Find the mode for: Height in cm 15 27 23 21 19 17 Number of Plants 1 2 6 12 7 5 1. 5 2. 6 3. 21 4. 10 15. The sales in a shop of men‘s shoes of different sizes in a month are given below Size (in inches) 7 7.5 8 8.5 9 9.5 10 10.5 11 Number of shoes sold 5 9 20 32 28 43 24 11 6 Find the mode 1. 43 2. 9.5 3. 9 4. 28 16. Given that the mean and mode of a unimodal distribution are 24.5 and 24.125 respectively. Find the median. 1. 21 2. 20.5 3. 24.375 4. 24.325 17. The Arithmetic mean and median of a unimodal distribution are 39 and 38 respectively .find the mode. 1. 24 2. 36 3. 34 4. 27 18. The range of the data set 15, 8, 3, 2, 11, and 12 is 1. 6 2. 13 3. 7 4. 16 19. Find the range of 35, 39, 40, 41, 42, 45, 40, 70, 65, 71, 36, 55, 62, 61. 1. 46 2. 36 3. 45 4. 37 20. Standard Deviation of the data set 0, 1, 2, 3, 4 is 1. 1.414 2. 1/\2 3. 2 4. 6.25 21. Find the standard deviation of 10, 20, 30, 40 and 50. 1. 200 2. 14.14 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 126 - 3. 11.14 4. None of these 22. Find the variance of 14, 8, 16, 12, 10. 1. 2.82 2. 8 3. 6 4. 1.41 23. Find the variance of 1, 3, 6, 5, 10. 1. 9.2 2. 5.7 3. 11.1 4. None of these 24. If D is the sum of the squares of the deviations form the mean of N observations, then its standard deviation is 1. D/N 2. \D/N 3. D/\N 4. \D/N 25. The standard deviation of a data set is 4. then its variance is 1. 1 2. 16 3. 8 4. \2 26. In a given distribution, the standard deviation is ‗d‘. Each data element in the distribution is doubled. What is the value of the standard deviation of the new data set? 1. d 2. 2d 3. d\2 4. d/\2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 127 - PROGRESSIONS ____________________________________________________________________________________ Exercise-1 1. A series in which every term beginning from the second is obtained by adding a fixed number to the term preceding it, is called a _________ a) Geometric Progression b) Arithmetic Progression c) Harmonic Progression d) A series in general 2. A series in which the nth and (n - 1)th terms are denoted by t n and t n-1 is called an Arithmetic Progression if for all n ≥ 2, a) t n -t n-1 is a constant b) tn t t n n 1 1 ÷ ÷ ÷ is a constant c) t t n n 1 1 1 ÷ ÷ is a constant d) t t n n 1 ÷ + is a constant 3. Which of the following series form an Arithmetic Progression? a) 2, 1, 2/3, ½ b) 2, 1, 0, -1 c) 2, 1, ½, ¼ d) 1, 1, 2, 3 4. What is the 12 th term of the sequence whose nth term is 6 ) 1 2 ( ) 1 ( + + n n n ? a) 364 b) 13 c) 25 d) 650 5. If t n of a series is given by t n = (n – 1) (n – 2), the second term is __________ a) 1 b) 2 c) 3 d) 0 6. The nth term of a series is given by an + b; then the 1 st term is ___________ a) 1 b) a c) b d) a + b 7. The nth term of an A.P. is given to be an + b; the common difference is _______ a) a b) b c) a + b d) a – b7 The nth term of a series is given to be 3n – 1. What kind of progression is it? a) A.P. b) G.P. c) H.P. d) None of these 8. The nth term of a series is given to be 3n – 1. What kind of progression is it? From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 128 - a) A.P. b) G.P. c) H.P. d) None of these 9. If tn of a series is given by 5 – 3n, it is a __________ a) A.P. b) G.P. c) H.P. d) None of these 10. The nth term of an A.P. whose initial term is 1 and the common difference is 3, is _______ a) 1 + n b) 1 + 3n c) 3n – 2 d) 3n - 1 11. If the 1 st and the 5 th terms of an A.P. are respectively 5 and 17, the common difference is _______ a) 2.4 b) 3 c) 4 d) 2 12. The 5 th term and the 17 the terms of an A.P. are 13 and 37 respectively. Then the 1 st term is _______ a) 3 b) 5 c) 7 d) 8 13. The 15 th term of the series 3, 5, 7, 9,…is ______ a) 29 b) 31 c) 33 d) 45 14. The 12 th term of the series 5, 14, 23 _____ is a) 239 b) 100 c) 104 d) 900 15. What is the 10 th term of the series 21, 19, 17, 15 ________ a) 1 b) 3 c) 11 d) -1 16. The nth term of the A.P. with initial term a and common difference d is given by t n = a) a + nd b) a + (n – 1)d c) a – nd d) an + d 17. The 1 st term of an A.P. is 48 and this common difference is -2. Which term of the series is „0‟? a) 24 b) 26 c) 25 d) 20 18. If a, x, b are in A.P., then x is called the ________ of a and b. a) Geometric mean b) Harmonic mean c) Arithmetic mean d) The second term 19. If x is the Arithmetic Mean of a & b, then x = From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 129 - a) 2 b a + b) 2 a b ÷ c) ab d) b a ab + 2 20. In the series 3, 7, 11, 15, ______, the first term that exceeds 100 is the ____th term. a) 25 b) 26 c) 24 d) 30 21. If 3 A.M.‟s are inserted between -4 and 4, the common difference of the series formed I ________ a) -2 b) 2 c) 3 d) 1 22. Common difference of the series in A.P. 13, 8, 3, -2, ___ is ___ a) 5 b) 10 c) -4 d) -5 23. The A.M. between ½ and 1/3 is ______ a) 5/6 b) 5/12 c) 12/5 d) 6/5 24. The tenth term of the A.P. 13, 8, 3, -2 _____ is a) -32 b) 32 c) -57 d) -52 25. There are n arithmetic means between and b. Their common difference is a) 1 + + n a b b) a b n + +1 c) 1 + ÷ n a b d) 1 + n ab From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 130 - Exercise-2 1. The sum of n A.M.s inserted between a and b is _______ a) (a + b) n b) 2 n (a + b) c) 2 n (b - a) d) 2n (a + b) 2. The first term of an A.P. is a; Common difference is d; sum to n terms denoted by S n = _______ a) 2a + (n – 1) d b) 2 n [a + (n – 1)d] c) 2 n [2a + (n – 1)d] d) [a + (n – 1)d] 3. There are n terms in an A.P., the first term is a, the last term is I. Their sum is _______ a) n (a + 1) b) n (a – a) c) 2 n [a + 1] d) 2 n [1 - a] 4. The first and the last terms of an A.P. are 3 and 39 respectively. If the number of terms is n, then S n = ____ a) n (39 + 3) b) 21n c) 18n d) 36n 5. Which of the following is written under the formula ¿ + ? 2 ) 1 (n n a) 1 2 + 2 2 + 3 2 + ….. + n 2 b) 1 3 + 2 3 + 3 3 + ….. + n 3 c) 1 + 3 + 6 + 10 + ….. d) 1 + 2 + 3 + 4 + …. n 6. The sum of the 1 st 10 terms of an A.P. = 4 times the sum of the first 5 terms‟ then the ratio of the 1 st term to the common difference is _____ a) 1 : 2 b) 2 : 1 c) 1 : 4 d) 4 : 1 7. The pth term of an A.P. is q and the qth term is p. Then the common difference is ______ a) 1 b) p – q c) q – p d) -1 8. The pth term of an A.P. is q and the qth term is p. Then the 1 st term is _______ a) p + q – 1 b) p + q + 1 c) q – p – 1 d) q – p + 1 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 131 - 9. The mth term of an A.P. is n and the nth term is m. Then the (m + n)th term is ________ a) m + n b) m – n c) n – m d) 0 10. A series in which every term beginning from the second bears a constant ratio to its predecessor, then the series is said to be a ________ a) A.P. b) G.P. c) H.P. d) A general series 11. If a is the 1 st term of a G.P. and r is the common ratio, the nth term denoted by t n is given by t n = _______ a) a n r b) ar n c) ar n-1 d) a.r(n – 1) 12. If the nth term of a series is 5. (-3) n , then it is a _______ a) G.P. b) H.P. c) A.P. d) None of these 13. The nth term of a G.P. is 3.2 n-1 . Then the common ratio is ________ a) 3 b) 2 c) ½ d) 5 14. The second term of a G.P. is 2 and its 6 th term is 32; the common ratio is ________ a) 2 b) 6 c) -2 d) 2 or -2 15. What is the nth term of the sequence 4, 12, 36, ______ ? a) 4.3 n b) 4 n-1 .3 c) 4.3 n-1 d) 4.3 n+1 16. The 2 nd and the 5 th terms of a G.P. are 2 and ¼. Then the common ratio is a) ½ b) -1/2 c) ½ or -1/2 d) -2 17. If a, b, c are in G.P., the a/b = a) b/c b) b/a c) a/c d) c/b 18. The common ratio of the series + + + 2 3 1 3 1 1 _______ is ________ a) 1/3 b) -1/3 c) -2/3 d) 3 19. The common ratio of the series 3-6+12-24+48 ______ is a) 12 b) -1/2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 132 - c) -2 d) -3 20. The common ratio of the series 4, , 2 , 2 , 2 2 ______ is _______ a) 2 b) 2 1 c) 2 1 ÷ d) 2 1 21. If the geometric mean between a and b is x; then x = _______ a) 2 b a + b) 2 ab c) ab d) ab 22. If a, b, c, are in G.P., then _______ a) a 2 = bc b) b 2 = ac c) c 2 = ab d) 2 c a b + = 23. If n positive geometric means are inserted between two positive numbers a and b, then their common ratio is ________ a) 1 + ÷ = n a b b b) 1 + ÷ n a b c) 1 + n a b d) 1 + + n b a 24. The product of n geometric means inserted between a and b is ______ a) (ab) n-1 b) (ab) n+1 c) (ab) n/2 d) ab n/2 25. If S n is the sum of n terms of the G.P. a, ar, ar 2 , ______, then S n = a) 1 ) 1 ( ÷ ÷ r r a n b) a (r n-1 -1) c) 1 1 ÷ ÷ r ar n d) 1 ÷ r ar n From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 133 - MATRICES ____________________________________________________________________________________ Exercise-1 1. A matrix is _________ a) a set of numbers b) an arrangement of numbers in a column c) an arrangement of numbers in a column d) an arrangement of numbers in rows and columns such that there are equal number of elements in each row and equal number of elements in each column 2. A matrix is said to be of order m x n if it has _______ a) m and n as its elements b) mn elements c) m rows and n columns d) m columns and n rows 3. Which of the following is a matrix of the order 2 x 3? a) | | . | \ | 2 1 1 2 b) | | . | \ | 3 2 3 1 3 2 c) | | | . | \ | 3 1 2 3 3 2 d) ( ) c b a 4. A matrix is said to be rectangular if it has a) more columns than rows b) more rows than columns c) unequal number of rows and columns d) as many columns as rows 5. A row matrix is matrix which contains a) only rows and no columns b) only one row c) unequal number of rows and columns d) as many columns as rows 6. A column matrix is a matrix which has a) only columns and no rows b) more columns than rows c) all elements of a column are zeroes d) only one column 7. An example of a row matrix is a) | | | . | \ | 3 2 1 b) ( ) 8 5 2 c) | | . | \ | 7 2 3 3 5 2 d) | | | . | \ | 7 7 2 5 3 2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 134 - 8. An example of a column matrix is __________ a) ( ) 8 5 1 b) | | | . | \ | 3 7 2 5 1 3 c) | | | . | \ | 8 7 3 d) | | | . | \ | 3 5 0 2 3 0 1 2 0 9. Which of the following may be the order of a row matrix? a) 2 x 1 b) 1 x 3 c) 2 x 2 d) 2 x 5 10. Which of the following cannot be the order of a row matrix? a) 1 x 3 b) 1 x 5 c) 1 x 4 d) 2 x 1 11. Which of the following can be the order of a column matrix? a) 1 x 3 b) 3 x 1 c) 4 x 2 d) 2 x 2 12. Which of the following is a rectangular matrix? a) | | . | \ | 0 3 2 3 2 1 b) | | . | \ | 3 4 4 3 c) | | | . | \ | 6 1 5 1 2 4 5 4 3 d) | | . | \ | 2 0 5 1 13. A matrix is said to be a square matrix if it has ________ a) all its elements perfect squares b) equal number of columns and rows c) if all elements of a row are equal d) if all its elements are equal 14. The example for a square matrix is ______ a) ( ) 3 2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 135 - b) | | . | \ | 2 3 c) | | | . | \ | 3 2 1 d) | | . | \ | 0 0 1 1 15. Two matrices are said to be equal if a) they contain exactly the same elements b) they have an equal number of elements c) if they are of the same order d) if they are of the same order and the elements are equal to the corresponding elements of the other, each to each. 16. if | | . | \ | = | | . | \ | ÷ + 8 2 5 8 2 2 5 3 y x a) x = 11; y = 6 b) x = 11 y = 10 c) x = 5; y = 10 d) x = 5; y = 6 17. A zero matrix is a matrix which has ________ a) no elements at all b) all elements of a column are zeroes c) all its elements zeroes d) all elements of a row zeroes 18. A is matrix; the matrix obtained by interchanging its rows and columns is known as ______ of A. a) inverse of A b) transpose of A c) square of A d) none of these 19. The transpose of A is denoted by ________ a) –A b) A -1 c) A T d) AT 20. Which of the following is the transpose of A if | | | . | \ | = 4 5 2 1 3 2 A a) | | | . | \ | 5 1 1 2 2 3 b) | | . | \ | 4 2 3 5 1 2 c) | | . | \ | 5 1 2 4 2 3 d) | | . | \ | 4 5 2 1 3 2 21. One of the following matrices has its transpose = the matrix itself. Which is it? From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 136 - a) | | | . | \ | 3 2 5 2 4 2 5 2 3 b) | | . | \ | 6 5 4 3 2 1 c) | | | . | \ | 6 5 4 3 2 1 d) | | | . | \ | 5 6 3 4 1 2 22. Matrix A is of the order 3 x 2; the order of A T is _________ a) 3 x 2 b) 3 x 3 c) 2 x 2 d) 2 x 3 23. | | . | \ | = 5 4 3 2 A ; the transpose of the transpose of A = a) | | . | \ | 3 4 5 2 b) | | . | \ | 4 3 5 2 c) | | . | \ | 5 3 4 2 d) | | . | \ | 5 4 3 2 24. If A and A T are of the same order, then A must be a) a column matrix b) a row matrix c) a square matrix d) a null matrix 25. A is a square matrix of order 3 x 3; A T is its transpose. Then ________ a) the elements of the principal diagonal of A are equal to the corresponding elements of the principal diagonal of A T b) the elements of the two diagonals of A are equal to the corresponding elements of those of A T c) the elements at the ends of rows are equal in both. d) None of these. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 137 - Exercise-2 1. A scalar matrix is a square matrix in which a) all elements zeroes b) all elements of the principal diagonal are equal non-zero numbers c) all elements of the principal diagonal are equal non-zero numbers and the others are zeroes. d) All elements of the principal diagonal are equal non-zeroes. 2. Which of the following is a scalar matrix? a) | | . | \ | 5 2 1 5 1 2 b) | | | . | \ | 5 5 2 1 1 2 c) | | | . | \ | 2 0 0 0 2 0 0 0 2 d) | | | . | \ | 0 0 2 0 2 0 2 0 0 3. A unit matrix is a square matrix in which a) all elements are ones b) all elements of the principal diagonal are ones c) all elements of the principal diagonal are zeros and the others are all non-zero numbers d) all elements of the principal diagonal are ones and the other elements are zeros 4. Which of the following is a unit matrix? a) | | . | \ | 2 0 0 2 b) | | . | \ | 0 1 1 0 c) | | . | \ | 1 1 1 1 d) | | . | \ | 1 0 0 1 5. Addition of the two matrices is defined only if ________ a) the two matrices are square matrices b) the two matrices are of the some order c) one of them is the transpose of the other d) both are rectangular matrices or both square matrices 6. The addition of A and A T is defined if _________ a) A is null matrix b) A is column matrix c) A is row matrix d) A is square matrix From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 138 - 7. A = | | . | \ | 1 2 5 2 1 3 , B = | | . | \ | 0 0 1 1 0 0 ; then A + B = ________ a) | | . | \ | 1 2 6 3 1 3 b) | | . | \ | 0 0 5 2 0 0 c) | | | . | \ | 1 2 2 1 6 3 d) not defined 8. A and B are two matrices on which addition if defined. A + B = B + A = A. Then _______ a) A is zero matrix b) B is zero matrix c) A and B are zero matrices d) A is a unit matrix 9. A and B are two matrices such that A + B = 0 2x3. Then B is called _______ a) The transpose of A b) The inverse of A c) The additive inverse of A d) None of these 10. The additive inverse of A if A = | | . | \ | 4 2 5 3 is a) | | . | \ | 0 0 0 0 b) | | . | \ |÷ 4 2 5 3 c) | | . | \ | ÷ ÷ 4 2 5 3 d) | | . | \ | ÷ ÷ ÷ ÷ 4 2 5 3 11. If A = | | . | \ | 7 6 5 4 ; B = | | . | \ | 5 4 3 2 , then A-B = __________ a) | | . | \ | 7 6 5 2 b) | | . | \ | 2 2 2 2 c) | | . | \ | 2 7 5 d) | | . | \ | 3 2 5 4 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 139 - 12. A = | | . | \ | 5 4 4 3 and B = unit matrix of order 2 x 2; then A + B = _______ a) | | . | \ | 7 6 5 4 b) | | . | \ | 6 4 4 4 c) | | . | \ | 5 5 5 3 d) | | . | \ | 5 4 4 3 13. P = | | . | \ | 4 3 2 1 , then 5P = ________ a) | | . | \ | 4 3 10 5 b) | | . | \ | 20 15 2 1 c) | | . | \ | 20 15 10 5 d) | | . | \ | 20 3 2 5 14. If A = | | . | \ | 0 2 8 4 and 3A – 2B = 0 then B = ________ a) | | . | \ | 0 1 5 2 b) | | . | \ | 0 3 12 6 c) | | . | \ | 0 2 9 6 d) | | . | \ | 0 2 12 6 15. Product of two matrices A and B, denoted by AB is defied only when ________ a) both are square matrices of the some order b) B is the transpose of A c) when B has as many columns as A has rows d) when B has many rows as A has columns 16. If A is of order m x n and B is of order n x P. then AB is of order _______ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 140 - a) m x n b) n x p c) m x p d) p x m 17. A = | | . | \ | 4 2 1 5 3 2 ; B is one of the following matrices, such that AB is defined Then B = _________ a) | | . | \ | 64 5 4 3 2 1 b) | | | . | \ | 6 3 5 2 4 1 c) ( ) 3 2 1 d) | | . | \ | 0 1 2 2 18. A is a matrix of order 3 x 2; B is a matrix such that AB is defined. Which of the following can the order of B be? a) 3 x 2 b) 2 x 2 c) 3 x 1 d) 1 x 3 19. A is of order of 2 x 1 and B is order 1 x 2; the order of AB = ______ a) 1 x 1 b) 2 x 1 c) 2 x 2 d) 1 x 2 20. A is of order 1 x 2 and AB is of order 1 x 3; then B is of order ________ a) 1 x 2 b) 2 x 3 c) 1 x 3 d) 3 x 1 21. AB is of orde 2 x 3; if B also is of order 2 x 3 then A is of order a) 2 x 3 b) 2 x 2 c) 3 x 2 d) 3 x 3 22. If A = | | | . | \ | 6 5 4 3 1 2 and B = | | | . | \ | 5 2 1 then AB = ________ a) | | | . | \ | 5 6 2 b) | | | . | \ | 18 8 1 c) | | | . | \ | 13 4 3 d) does not exist From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 141 - 23. A = | | . | \ | 4 3 2 1 ; B = | | . | \ | 0 1 2 3 ; then AB = _______ a) | | . | \ | 0 3 4 3 b) | | . | \ | 4 4 4 4 c) | | . | \ | 2 1 14 9 d) | | . | \ | 6 13 2 5 24. The product of A and A T is always a a) unit matrix b) a square matrix c) a rectangular matrix d) a scalar matrix 25. Give A B, C, O are matrices of the same order, only one of the following I false. What is it? a) A + B = B + A b) (A + B) + C = A + (B + C) c) O + A = A + O = A d) AB = BA From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 142 - STATEMENTS ____________________________________________________________________________________ Exercise-1 1. A sentence which if either true or false but not both is called a) open sentence b) statement c) expression d) negation 2. Which of the following is a statement? a) x + 4 = 7 b) What is 3 + 4 ? c) x 2 = 1 d) x > 5 3. Which of the following sentence is a statement? a) 7 > 9 b) Is 7 + 3 < 7 + 2? c) x + 7 d) What is 7 + 3? 4. The denial of a statement is called a) conjunction b) disjunction c) implication d) negation 5. ______ is used to join two or more statements a) Connective b) Quantifier c) Statement d) none 6. If two or more sentences are joined by a connective, it is called a) compound statement b) conjunction c) disjunction d) biconditional 7. A compound statement p or q is called a) disjunction b) conjunction c) implication d) bi-implication 8. A compound statement p and q is called a) conjunction b) disjunction c) conditional d) bi-conditional 9. A compound statement if p then q is called a) conjunction b) implication c) disjunction d) bi-implication 10. The connective for double implication is a) . b) v c) · d) ¬ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 143 - 11. A statement having the connective “If and only is” is called a) conjunction b) bi-conditional c) implication d) disjunction 12. The symbol for implication is a) ÷ b) ¬ c) . d) ÷ 13. Symbol for bi-implication is a) ¬ b) · c) . d) v 14. The symbol for „and‟ is : a) . b) v c) ¬ d) · 15. The connective in a disjunction is: a) and b) or c) if……then d) if and only if 16. Symbol for „either p or not p‟ is : a) p v ~p b) ~p c) p ^ ~p d) p ¬ ~p 17. p . q is true only when a) p is true b) q is true c) both p and q are false d) both p and q are true 18. p ¬ q is true only when a) p v ~p b) ~p c) p ^ ~p d) p ¬ ~p 19. If p ¬ q is false when a) ~q ¬ ~p b) ~p ¬ ~q c) q ¬ p d) ~p ¬ q 20. If p is false and q is true, which of the following is false a) ~p b) ~q c) ~p ÷ q d) ~(p ÷ q) 21. If p is false, the truth value of ~p is a) F b) T c) F and T d) F or T From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 144 - 22. p : x + y = 5; q = x – y = 1. The (x, y) which make p . q true is : a) (5, 1) b) (1, 5) c) (3, 2) d) (2, 3) 23. If p is a false statement, the truth value of p ¬ q is : a) T b) F c) Both T and F d) T or F 24. The truth value of p . ~p is _____ a) F b) T c) F or T d) Cannot be decided 25. If p is a true statement and q is a false statement, then the truth value of p . q is a) True b) False c) Both true and False d) Either true nor false From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 145 - Exercise-2 1. If p is any non-empty set and C is the null set, then the truth value of p . q is a) C b) p c) µ d) p 1 2. If p is a true statement and q is a false statement, then the truth value of p v q is a) True b) False c) Both true and false d) Neither true nor false 3. If p is 5x – 10 = 0; q is x = 2, then p ÷ q is a) True b) False c) Both true and false d) Neither true nor false 4. Which of the following is always false, except in the case of both p and q are true? a) p ÷ q b) q ÷ p c) p . q d) p v q 5. The truth value of p ¬ q is T. The truth value of ~q ¬ ~p is : a) T b) F c) T v F d) T . F 6. p : 3 > 1; q : 8 < 10, then p . q is a) False b) True c) True or False d) None of these 7. The negation of the statement “The earth is a planet” is a) The earth is not a planet b) The earth is not round c) The earth is round d) The earth moves around the sun 8. The negation of “all primes are odd” a) some primes are odd b) only some primes are odd c) some primes are even d) some primes are not odd 9. T the negation of p . q is : a) ~p . ~q b) p v q c) ~p v ~q d) q . p 10. The negation of p v q is : a) p . q b) ~p v ~q c) ~p . ~q d) q v p From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 146 - 11. The negation of p ¬ q is : a) q ¬ p b) ~p ¬ ~q c) p v ~q d) p . ~q 12. The negation of p · q is : a) q · p b) ~p · ~q c) ~p · q d) q · p 13. The negation of „2 + 3 = 5 . 4 x 3 = 12‟ is a) 2 + 3 = 5 . 4 x 3 = 12 b) 2 + 3 = 5 v 4 x 3 = 12 c) 2 + 3 = 5 . 4 x 3 = 12 d) 2 + 3 = 5 v 4 x 3 = 12 14. The negation of the statement, “If two lines are intersecting each other, then they are not parallel” is a) If two straight lines are not intersecting each other, then they are not paralled. b) If two straight lines are intersecting each other, then they are not parallel. c) If two straight lines are not intersecting each other, then they are parallel. d) If two straight lines are intersecting each other, then they are parallel. 15. Which of the following is a true statement? a) 2 + 3 = 7 . 3 + 4 = 7 b) 3 + 4 = 7 . 4 + 5= 12 c) 2 + 3 = 5 . 2 + 4 = 6 d) 3 + 4 = 7 . 4 + 3 = 12 16. Which of the following is a false statement? a) 2 + 3 = 5 v 2 x 3 = 6 b) 3 + 2 = 5 v 3 x 4 = 7 c) 4 + 3 = 8 v 3 x 4 = 12 d) 5 + 2 = 8 v 3 x 2 = 7 17. True statement of the following bi implications is a) 3 + 7 = 10 ÷ 1 + 2 = 2 b) 3 + 21 = 10 ÷ 1 + 2 = 3 c) 3 + 7 = 10 ÷ 1 + 2 = 3 d) 3 x 7 = 10 ÷ 1 + 2 = 3 18. If which of the following implication is false? a) If 3 + 8 = 11, then 1 x 0 = 0 b) If 3 + 8 = 11, then 1 x 0 = 1 c) If 3 + 8 = 10, then 1 x 0 = 0 d) If 3 + 8 = 10, then 1 x 0 = 1 19. Which of the following is false? a) 3 x 2 = 6 ÷ 4 x 1 = 4 b) 3 x 2 = 6 ÷ 5 x 0 = 5 c) 1 x 4 = 5 ÷ 2 x 3 = 5 d) 4 x 1 = 5 ÷ 4 x 0 = 0 20. True statement of the following bi-implications is … a) 5 + 4 = 9 ÷ 3 + 2 = 6 b) 3 + 2 = 6 ÷ 3 x 2 = 6 c) 5 + 4 = 9 ÷ 5 x 4 = 20 d) 5 x 4 = 9 ÷ 5 + 4 = 9 21. Which of the following is true? From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 147 - a) 2 + 4 = 5 v 4 + 3 = 5 b) 3 x 5 = 8 . 2 x 3 = 6 c) x 2 = 4 ¬ x = 2 d) 3 x 7 = 10 · 1 x 2 = 3 22. True statement of the following a) 4 + 5 = 9 . 2 x 0 = 20 b) 4 x 5 = 9 . 2 x 0 = 0 c) 4 – 5 = 1 . 4 x 5 = 20 d) 4 + 5 = 9 . 2 x 0 = 0 23. Which of the following is a true statement? a) (a, b) = (b, a) b) a e (a, b) c) {(a, b)} e (a, b) d) (a, b) e {(a, b)} 24. Which of the following is a universal statement? a) Some roses are red b) No rose is is red c) There exists atleast one rose which is red d) A few roses are red 25. If two compound statements have the some truth values, they are ______ a) Congruent statements b) Conditionals c) Disjounctions d) None From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 148 - SETS ____________________________________________________________________________________ Exercise-1 1. Which of the following collections is well-defined ? And hence, a set is a) A collection of clever students in a class b) A collection of tall boys in a class c) A collection of all natural numbers d) A collection of some even numbers 2. The set of natural numbers less than 5 is a) {1, 2, 3, 4, 5} b) {1, 2, 3, 4} c) {0, 2, 4} d) {0, 1, 2, 3, 4} 3. The roster form of A = {x/x = x} is a) {1} b) {0} c) {0, 1} d) { } 4. B = {x/x e N; x is even; x lies between 2 and 10}. In Roster form it is …. a) B = {2, 4, 6} b) {4, 6, 8} c) {4, 8} d) {2, 4, 6, 8, 10} 5. A = {x/x e N; 1 < x < 5} is equal to a) {2, 3, 4} b) {1, 5} c) {0, 1, 3, 5} d) {1, 2, 3, 4, 5} 6. {1, 4, 9, 16, …..} in set builder form is ………. a) {x /x = y 2 , y e Z} b) {x / x = y, y e Z} c) {x / x = y 2 , y e Z} d) {x / x 2 = 2/y, y e Z} 7. {1, ½, 1/3, 1/4, …….} in set builder form is ……… a) {x / x = 1/y, y e N} b) {x / x = y, y e N} c) {x / x = 1/x, x e N} d) {x / x = 1/y, y e R} 8. If A = {1, 0, 2, 3}, B = {2, 4, 3, 5}, then A B = a) {0, 1, 2, 3, 4, 5} b) {2, 3} c) {1, 0} d) {4, 5} 9. If A = {0, 1, 3, 5, 8}, B = {0, 3, 6, 5, 9} then A · B = a) {0, 1, 3, 5, 6, 8, 9} b) {0, 1, 3, 6, 9} c) {0, 3, 5} d) {1, 8, 6, 9} 10. If A = {2, 3, 5, 7, 8}, B = {3, 5, 4, 9}, A – B = a) {2, 7, 8} b) {2, 7, 8, 9} c) {4, 7, 8, 9} d) {3, 5} 11. If µ = {1, 2, 3, 4, 5}, A = {3, 5}, then A‟ = From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 149 - a) {1, 2, 4} b) {3, 4} c) {1, 2, 3, 4, 5} d) C 12. If X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and Y = {1, 3, 5, 7}, then X · Y is a) Y b) X c) ¢ d) µ 13. A = {1, 2, 4, 6, 8}; B = {3, 5, 7}; C = {9, 10}; then A B C = a) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} b) C c) {1, 2, 3, 5, 7, 10} d) {3, 5, 7, 9, 10} 14. If A = {1, 2, 3, 5, 6} and B = {2, 3, 4, 7} then A – B = a) {2, 3} b) {1, 5, 6} c) {4, 7} d) {2, 3} 15. If x = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; y = {1, 3, 5, 7}, x – y = ______ a) y b) x c) C d) {2, 4, 6, 8, 9, 10} 16. W is the set of whole numbers and N is the set of Natural numbers; then W – N = a) C b) 0 c) {0} d) µ 17. A = {1, 2, 3}; B = {3, 5}, then A x B = _______ a) {(1, 1), (1, 2), (1, 3), (1, 5), (2, 3)} b) {(1, 3), (1, 5), (2, 3), (2, 1), (2, 3)} c) {(1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5)} d) {(3, 1), (3, 2), (3, 5), (5, 1), (5, 2), (5, 5)} 18. If A = {1, {2, 3} 4, 5} then which of the following is true? a) {2, 3} c A b) I c A c) 3 e A d) {2, 3} e A 19. {x/ = -2 < x < 2, x e z} = ………… a) {-2, -1, -0, 1, 2} b) {-1, 0, 1} c) {-1, 0, 1, 2} d) {-1, 1, 2,} 20. If A = {1, 2, 3}, B = {4, 5, 6}, then A‟ B‟ = …….. a) Cannot be decided since µ is not given b) | c) µ d) None 21. A = {1, 2, 3}, B = {4, 5, 6}, then (A · B)‟ = a) cannot be decided since µ is not given b) | c) µ d) none of the above three 22. A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. B = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} then n (A A B) = …. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 150 - a) 20 b) 10 c) 100 d) {2, 4, 6, 8, 10} 23. Which of the following is a null set? a) {x / x is a female prime minister of India} b) {x / x = x} c) {x / x is a vowel in English alphabets} d) {x / x is a letter before „a‟ in English alphabet} 24. One of the following is an infinite set. a) The set of all points in the line segment AB b) Set of all od numbers below one crore c) Set of prime numbers d) Set of people now living in the world 25. Which of the following is a finite set? a) {Multiples 48} b) Factors of 48 c) Natural numbers d) Rational numbers between 1 and 2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 151 - Exercise-2 1. Which of the following is an example of null set? a) Set of all natural numbers between 13 and 17 b) Set of all rational numbers between 13 and 17 c) Set of primes between 13 and 17 d) Set of even numbers between 13 and 17. 2. Which of the following is an example of an infinite set? a) The set of people now living in the world. b) The set of stars in the sky. c) The set of all points in a chord AB d) Set of all even numbers below one crore 3. „A is a proper subset of B‟. a) B has more elements than A b) B has all the elements of A and at least one more c) A has all the elements of B at least one more than B d) A = B 4. Which of the following is a finite set? a) The set of real number b) The set of Rational numbers c) The set of irrational numbers d) The set of all the people in the world 5. Which of the following is the Null set? a) {x : x > 100, x e N} b) {x : x < 2, x e Z} c) {x : x < -10, x e Z} d) {x : x < -2, x e N} 6. To Which of the following sets does 2 belong? a) N b) Z‟ c) Q‟ d) Q 1 7. {x / x is square root of Negative integer, x e R} is ……… a) A universal set b) The Null set c) Infinite set d) None of the above 8. AB = CD an AB · CD = |. Then ABCD is a ........ a) Trepazium b) Parallelogram c) Kite d) None of the above 9. A _ B ¬ ________ a) A has fewer elements than B b) If x e B, x e A c) If x e A, x e B d) x belongs to both or does not belong to both 10. (A B)‟ = a) A‟ · B‟ b) A‟ B‟ c) A‟ c B‟ d) B‟ c A‟ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 152 - 11. If A and B are subsets of µ, A · B‟ = a) A‟ B b) A - B c) µ - B d) µ 12. If A c B; then A · B = a) B b) A · B c) A – B d) A 13. (A B) = B, then a) B c A b) A c B c) A = B d) A B 14. A and B are two sets – If n(A) = n(A) = n(B), then they are a) equal sets b) equivalent sets c) They have exactly some elements d) Each is a subset of the other 15. If A c B and B c A, then a) A = B b) A ÷ B c) A B = C d) A · B = C 16. A – B = a) {x/x e A and x e B} b) {x/x e A and x e B} c) {x/x e A and x e B} d) {x/x e A and x e B} 17. If A and B are disjoint sets then A · B = a) C b) A c) B d) none of these 18. A and B are two n on-empty sets and A · B = C then A – B = a) A b) B c) C d) none of these 19. The complement of µ is a) A b) ¢ c) µ d) A 20. A B = a) {x/x e A . x e B} b) {x/x e A . x e B v x e A and x e B} c) {x/x e A . x e B} d) {x/x e A . x e B} 21. A _ B means From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 153 - a) If x e B, then x e A b) If x e A, then x e B c) A and B are equal sets d) A has fewer elements than B 22. A · (B C) = a) A · (B C) b) (A · B) (A · C) c) A · B · C d) (A · B) · (A C) 23. (A · B)‟ = a) A‟ b) B‟ c) A‟ · B‟ d) A‟ B‟ 24. If p is a set, then (p‟)‟ is a) ¢ b) µ c) p‟ d) p 25. If P is a non empty set and ¢ is the null set, then P · ¢ is a) ¢ b) p c) µ d) p‟ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 154 - REAL NUMBERS, RATIONAL NUMBERS & LAW OF INDICES ____________________________________________________________________________________ Exercise-1 1. The product of m factors (m e N) each of which is a, is denoted a) a m b) m 0 c) ma d) m/a 2. In 3a m , m is called ____ a) power of a b) index (exponent) of the power of a c) the base of the power of m d) Coefficeent of 3a 3. In n | . | \ | 3 2 , the base of the power is ____ a) 2 b) 3 c) n d) 2/3 4. a m x a n = ____ (m, n e Q) a) a mxn b) a 2mn c) a m+n d) a m-n 5. 2 2 x 4 3 = ______ a) 2 5 b) 4 5 c) 2 8 d) 2 12 6. (a m ) n = ______ a) mn a b) n m a c) n m a + d) n m a 7. m > n, m, n e N, and x m > x n if ______ a) x > 1 b) x = 1 c) 0 < x < 1 d) x < 0 8. 0 < x < 1; m, n e N; m > n; then a) x m > x n b) x m = x n c) x m < x n d) none of these 9. (ab) m = _____ a) ab m b) a m b c) a m b m d) abm From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 155 - 10. _____ = | . | \ | m b a a) a m /b b) a/b m c) a m /b m d) a/bm 11. x m x x n = X m+n ; then _____ a) m, n, e N b) m, n e Z c) m, n, e Q d) m, n, e R 12. a m-n x a n-m = _______ a) 1 b) 0 c) a (m-n)2 d) a m-n-m-n 13. x m-n x x n-p x x p-m = _____ a) 1 b) 0 c) x (m-n)(n-p)(p-m) d) x 2m-2n-2p 14. a 3/2 x a 1/2 x a 1/2 a) a 5/2 b) a 3/8 c) 3x 5/2 d) 2x 5/2 15. m p m p p n p n m n m x x x xn x x + + + | | . | \ | | . | \ | | | . | \ | a) 1 b) 0 c) ( ) ( ) ( ) | | . | \ | + + + + q p q p p n p n n m n m a d) none of these 16. a 1/p expressed in radical form = _______ a) p a b) a 1 c) p a d) ap 17. a p/q = _____ a) a p /q b) ap/q c) p q a d) q p a From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 156 - 18. _____ = m n a a) a m/n b) a n/m c) a n-m d) m.a 1/n 19. ____ = m n x a) m n x b) n m x c) mn x d) n m x 20. If 12 x x p q = then a) p/q = 12 b) pq = 12 c) q/p = 12 d) p – q = 12 21. 2 x x _____ = 2 x+3 a) 3 b) x x 3 + c) 2 3 d) 3 + x x 22. x m+k = x m – 3, then k = ______ a) 3 b) -3 c) x 3 d) x -3 23. (3x 2 ) 3 = ______ a) 27x 2 b) 27x 5 c) 27x 6 d) 3x 6 24. 5 -1/4 = _____ a) 4 5 1 b) 4 5 1 c) 4 5 ÷ d) 4 5 ÷ 25. 3 x = 81; then 3 x+2 = a) 83 b) 162 c) 729 d) 90 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 157 - Exercise-2 1. 5 x = 625; What is the value of 5 x-2 ? a) 623 b) 625/2 c) 25 d) 600 2. If 3 x = 8, what is the value of 3 2x ? a) 16 b) 64 c) 48 d) 24 3. 4 x = 1024; then 2 x = _____ a) 512 b) 32 c) 1022 d) Can‟t be found 4. 3 x+3 = 729; then 3 x-3 = _____ a) 0 b) 1 c) 27 d) 81 5. 81 1/4 = _____ a) 3 b) -3 c) 1/3 d) – 1/3 6. If x = 0.2, then x 3/2 _____ a) 0.8 b) 8 c) .08 d) .008 7. (0.0064) 1/2 = _______ a) .8 b) .008 c) .08 d) .0008 8. 16 1/4 x 8 -1/3 = ______ a) 3 1 4 1 2 ÷ b) 4 1 3 1 2 ÷ c) 1 d) 12 1 128 ÷ 9. _____ 4 3 = x x a) x b) x 1 c) 12 x d) 4 3 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 158 - 10. If x x x = ( ) x x x ; then x = a) 0 b) 1 c) 3/2 d) 3 11. 2 1 1 ÷ + x + 2 1 1 x + = a) 0 b) 1 c) x 2 d) + x 2 12. 3 2 64 ÷ = a) 4 b) ¼ c) 8 d) 1/16 13. If a 2 = 0.04; then a 3 = a) 0.02 b) 00.8 c) 0.08 d) 0.008 14. If x = 0.3, then x = a) 0.9 b) 0.009 c) 0.1 d) 0.09 15. ( )4 3 81 = a) 27 b) 1/27 c) -243/4 d) 4 3 81 ÷ 16. (0.027) 2/3 = a) 0.018 b) 0.009 c) 0.09 d) 0.006 17. = | . | \ | + | . | \ | 3 1 4 1 8 125 625 81 a) 4.2 b) 3.6 c) 3.5 d) 3.1 18. If x 3 = 125 27 ; then x 2 = a) 9/25 b) 3/5 c) 9/5 d) 9/125 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 159 - 19. If 2 x – 2 x-1 = 4; then x x = a) 1 b) 27 c) 3 d) 2 20. 3 2 a x 3 1 a = a) 9 2 a b) ( ) 3 1 3 2 2 + a c) a 2 d) a 21. x -m = (m e N) a) –m . x b) m x 1 c) x . m 1 d) x m 22. = ÷3 12 a a a) a 9 b) a -15 c) a 15 d) 1/a 15 23. a 0 (a = 0) = a) a b) 0 c) · d) 1 24. 2 x+3 is obtained by multiplying 2 x with a) 2 b) ½ c) 8 d) 1/8 25. If (3 a ) x = 3 4a , then x = a) 3 b) 4 c) 3a d) 4a From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 160 - SURDS ____________________________________________________________________________________ Exercise-1 1. A surd is ______ a) any irrational number b) irrational root of a rational number c) irrational root of a nutural number d) irrational root of an irrational number 2. n a is a surd if and only if _______ a) n a b) n is rational c) a is rational and n a is irrational d) a is rational 3. Which of the following number is a surd? a) 3 64 b) 3 16 c) 3 8 27 d) 5 32 4. Which of the following is not a surd? a) 8 b) 3 16 c) 4 8 d) 5 8 5. n a is a surd of order : a) a b) n c) n a d) a n 6. Which of the following is a quadratic surd? a) 3 2 a b) a c) 9 d) 3 4 7. Two surds x and y are said to be like if ______ a) x and y are natural numbers b) x/y is a natural number c) x/y is a rational number d) x/y is irrational 8. The product of two quadratic like surds is a From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 161 - a) like surd with them b) an irrational number c) a rational number d) a natural number 9. If the product of two surds is a rational number, each of them is called a ______ of the other. a) rational factor b) like surd c) rationalizing factor d) factor 10. The rationalizing factor of the sured x is ______ a) x b) 3 x c) 2 x d) x 11. The rationalizing factor of 3 2 a is _____ a) 3 2 a b) 2 a c) 3 a d) 2 2 a 12. The rationalizing factor of b a ÷ is ______ a) b a + b) a b + c) b a + d) b a ÷ 13. What is the rationalizing factor of y x 3 2 + ? a) y x + b) y x 3 2 ÷ c) y x 2 3 ÷ d) y x ÷ 14. What is the rationalizing factor of 3 3 b a ÷ ? a) 3 3 b a + b) 3 2 3 3 2 b ab a + + c) 3 2 3 2 b a + 15. Which is the rationalizing factor of a - b 2 a) b a 2 + b) b a 2 + c) b a+ 2 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 162 - d) b a÷ 2 16. Which is the rationalizing factor of 3 3 y x + ? a) 3 2 3 2 y x + b) 3 2 3 3 2 y xy x + + c) 3 2 3 2 y x ÷ d) 3 2 3 3 2 y xy x + ÷ 17. What is the rationalizing factor of 3 y x + ? a) 3 2 3 2 y y x x + ÷ b) 3 y x ÷ c) 3 2 y x ÷ d) 3 2 y x ÷ 18. Two quadratic Surds are said to be conjugate to each other if _______ a) Their sum is rational b) Their product is rational c) if their sum of product is rational d) Their sum and product are both rational 19. What is the surd conjugate to b a+ ? a) b a÷ ÷ b) b a + c) b a÷ d) b a÷ 2 20. What is the surd conjuagate to b a 3 2 + a) b a 2 3 ÷ b) b a 3 2 ÷ c) b a 3 3 ÷ d) b a÷ 2 21. 3 2 1 + = _____ a) 3 2÷ b) 2 3÷ c) 2 3+ d) 6 5 2+ 22. If the denominator of the surd 6 2 5 1 + is rationalized, the resulting fraction equals ______ a) 6 2 5÷ b) 1 c) 12 5 + From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 163 - d) 6 5 2+ 23. = ÷ ÷ b a b a a) b a + b) b a ÷ c) a + b d) a - b 24. ( ) = | | . | \ | ÷ + ÷ 3 1 3 1 y x y x a) 3 2 3 2 y x + b) 3 2 3 1 3 1 3 2 y y x x + + c) 3 2 3 2 y x ÷ d) 3 2 3 1 3 1 3 2 y y x x + + 25. = + ÷ b ab a 2 a) b a + b) b a ÷ c) b a÷ d) b a+ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 164 - EXERCISE-2 1. If 3 2 1 ; 3 2 1 ÷ = + = b a then ab = a) 1 b) 0 c) 3 2÷ d) 3 2+ 2. If 3 4 7 3 2 5 + + is written in the form of a-b 3 then value of b is a) 5 b) 7 c) 2 d) 6 3. 3 3 4 7 3 2 5 b a÷ = + + ; then a = a) 11 b) 12 c) 9 d) -11 4. Rationalising factor of 7 2 x is a) 5 7 x b) 7 x c) 7 5 x d) 5 5 x 5. If the sum and product of two surds is a rational number, then they are _______ surds. a) similar b) dissimilar c) conjugate d) none 6. If 2 3 2 3 ÷ + = x then = + x x 1 a) 3 b) 5 c) 2 d) 10 7. = + + + ÷ ÷ 3 5 3 3 5 1 3 2 1 a) 1 b) 0 c) -1 d) 2 8. The simplest form of is 3 768 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 165 - a) 3 12 4 b) 3 12 8 c) 3 12 16 d) none 9. = + + + ÷ + 2 3 6 2 6 2 4 3 6 2 3 a) 2 b) 5 2 c) 0 d) 1 10. = ÷ 6 12 30 a) 2 3 3 2 ÷ b) 3 2 2 3 ÷ c) 3 5 5 3 ÷ d) 5 6 ÷ 11. ( ) 6 2 5 2 3 ÷ + = a) 0 b) 1 c) -1 d) 2 12. = + 6 2 3 3 a) ( ) 1 2 3 4 1 + b) ( ) 1 3 2 4 1 + c) 1 3 3 + d) 1 3 2 + 13. = + 15 2 8 a) 5 3 ÷ b) 3 5 ÷ c) 5 3 + d) none 14. 5 3 5 3 ÷ + + a) 10 b) 8 c) 0 d) -2 15. = + + xy y x 4 4 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 166 - a) y x + 2 b) y x ÷ 2 c) x y 2 ÷ d) y x + 16. = ÷ + mn n m 2 a) n m+ b) m – n c) m + n d) n m÷ 17. = + + + 3 4 2 4 6 2 9 a) 6 5 2 + + b) 3 2 2 + + c) 8 3 + d) None 18. = + + ÷ 35 4 27 7 6 a) 2 5+ b) 2 5÷ c) 1 5+ d) 5 1÷ 19. = ÷ + ÷ 15 4 3 8 5 4 21 a) 5 2 3 2 ÷ + b) 5 3 3 ÷ c) 5 2 3 3 + + d) None 20. = ÷ ÷ + 7 6 16 7 6 16 7 a) a rational b) multiple of 7 c) 0 d) None 21. = + ÷ 2 48 68 12 a) 2 2÷ b) 2 2+ c) 2 4+ d) none From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 167 - 22. = ÷ 8 50 98 a) 0 b) -1 c) 1 d) 2 23. = + ÷ ÷ ÷ ÷ 84 10 1 60 8 1 140 12 1 a) 1 b) 2 c) -1 d) 0 24. The rationalizing factor of 3 81 a) 81 b) 3 81 c) 3 9 d) 9 25. 6 2 1 2 2 ÷ ÷ + ÷ x x x a) 3 2 ÷ ÷ + x x b) 3 2 ÷ + + x x c) 2 3 ÷ ÷ + x x d) 2 3 ÷ + + x x From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 168 - LINEAR EQUATIONS, INEQUATIONS & MODULUS ____________________________________________________________________________________ Exercise-1 1. The graph of 3x – 5y + 16 > 0 is _____ a) A line b) The half plane on the origin side of the line 3x – 5y + 16 = 0 c) The half plane on the side of the line not containing the origin d) The line 3x – 5y + 16 = 0 and the original side of it. 2. The graph of 2x – 3y + 5 ≥ 0 is _____ a) The region on the original side of the line 2x – 3y + 5 > 0 b) The line 2x -3y + 5 = 0 and the region on the origin side of it. c) The line 2x – 3y + 5 = 0 and the side of it not containing the origin d) The region on the side of the line not containing the origin. 3. The intersection of the graph of x – 2y + 3 ≥ 0 and the line x – 2y = 0 is _____ a) ø b) The line x – 2y + 3 = 0 c) The half plane on the origin side of the line x – 2y + 3 = 0 d) The half plane on the non-origin side of the line x – 2y + 3 = 0 4. The intersection of the graphs x + y – 5 > 0 and x + y – 5 < 0 is _____ a) The line x + y – 5 = 0 b) The half plane on the origin side of the line x + y – 5 = 0 c) ø d) The half plane on the non-origin side of the line x + y – 5 = 0 5. A point belonging to the region 5x – 4y – 9 > 0 is _____ a) (4, 5) b) (5, 4) c) (6, 4) d) (6, 7) 6. A point which satisfies 3x + y > 6 is …….. a) (1, 0) b) (2, -1) c) (1, 3) d) (2, 1) 7. A point in the region 2x – 3y < 5 is _____ a) (-1, -1) b) (3, -3) c) (-2, -4) d) (0, -4) 8. A point satisfying x + y ≤ 4 is _____ a) (2, 4) b) (3, 1) c) (3, 2) d) (3, 4) 9. The point which does not lie on the straight line x + y = 6 is _____ a) (3, 3) b) (2, 4) c) (3, 2) d) (4, 2) 10. A point which lies in the are represented by x + y > 3 is a) (1, 1) b) (1, 0) c) (1/3; 1/3) d) (1, 3) 11. A point which satisfies the condition x + 2y + 5 ≥ 0 is _____ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 169 - a) (2, -1) b) (1, -4) c) (2, -4) d) (-3, 0) 12. A point which lies in the area 3x + y – 5 ≤ 0 is _____ a) (1, 2) b) (1, 3) c) (3, -1) d) (4, 0) 13. Point (2, 3) lies in the area represented by a) x + y < 2 b) x + y > 5 c) 2x + y < 6 d) x + 2y > 7 14. (3, 4) belongs to the area represented by _____ a) 3x – 4y + 8 ≤ 0 b) 3x – 4y + 7 ≤ 0 c) 3x + 4y + 9 ≤ 0 d) 3x – 4y > 0 15. Which of the following points lies on the same side of the line 4x + 3y – 25 = 0 as (3, 4)? a) (4, 3) b) (3, 5) c) (4, 4) d) (2, 3) 16. Which of the following points lies on the side of 2x + 3y – 12 = 0 not containing (3, 3)? a) (1, 1) b) (2, 3) c) (1, 4) d) (-1, 5) 17. Which pair of the points lie on the same side of the line 5x – 2y + 5 = 0? a) (2, 1), (1, 2) b) (-1, 1), (3, 2) c) (5, 2), (-5, -2) d) (-1, -1), (-2, -1) 18. Which of the following pairs of points lie on different sides of the line 5x + 3y – 15 = 0. a) (1, 6), (2, 1) b) (2, 2), (1, 4) c) (0, 0), (-2, 2) d) (5, 3), 6, 2) 19. A point which lies on the some side of the line 3x + 5y – 30 = 0 containing the origin is _____ a) (0, 0), (1, 1) b) (2, 1), (0, 2) c) (1, 0), (0, 1) d) (-1, 0), (0, 0) 20. A point which lies on the same side of the line 3x + 5y – 30 = 0 containing the origin is _____ a) (10, 1) b) (8, 2) c) (7, -1) d) (7, 3) 21. 3x + 2y should not be less than 50. This can be represented as _____ a) 3x + 2y > 50 b) 3x + 5y = 50 c) 3x + 2y ≥ 50 d) 3x + 5y ≤ 50 22. 3x + 2y would be at most 50. This can be represented as _____ a) 3x + 2y < 50 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 170 - b) 3x + 2y ≤50 c) 3x + 5y > 50 d) 3x + 2y = 50 23. 3x should not exceed 5y by more than 10. This can be represented as _____ a) 3x – 5y ≤ 10 b) 3x – 5y = 10 c) 3x – 5y < 10 d) 3x – 5y > 10 24. A chair costs Rs. 50 and a table costs Rs. 150. The cost of x chairs and y tables should not exceed R. 5000. How do you represent this? a) 50x + 150 y = 5000 b) 50x - 150 y < 5000 c) 50x + 150 y ≥ 5000 d) 50x + 150 y ≤ 5000 25. If an iso-profit line coincides with a boundary of the feasible polygonal region, we have ____ a) 1 solution b) more than 1 solution c) Infinite number of solutions d) no solution From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 171 - EXERCISE-2 1. The objective function sought to be maximized is 3x + 4y. The vertices of the polygonal region representing the solution set are (0, 0), (0, 1), (8, 0), (4, 8). At which vertex has it the minimum value? a) (0, 0) b) (8, 0) c) (4, 8) d) (0, 1) 2. The objective function sought to be minimized is 5x + 2y. The vertices of the polygonal region representing the solution the solution set are (0, 16), (4, 18) and (12, 0). At which vertex has it the minimum value? a) (0, 16) b) (12, 0) c) (4, 8) d) none of these 3. The solution set which satisfies the inequations x 2 – 4x + 3 < 0 a) (1, 4) b) (1, -4) c) (1, -3) d) (-4, 3) 4. x < 0; y > 0; (x, y)lies in the quadrant a) Q 1 b) Q 2 c) Q 3 d) Q 4 5. The point which does not lie in the region 2x – 3y > 5. a) (1, 1) b) (3, -3) c) (-2, -4) d) (0, -4) 6. The inequations with the solution set 1 < x < 3 is a) x 2 + 4x + 3 > 0 b) x 2 – 4x + 3 < 0 c) x 2 – 4x – 3 = 0 d) x 2 – 4x + 4 > 0 7. The point which belongs to the region indicated by the inequations x + 3y < -5 is a) (2, 1) b) (-2, -1) c) (-3, 1) d) (-3, -1) 8. Which of the following inequations represents the region containing the points (1, 2) and (2, 1) a) x + y < 2 b) x + y > 5 c) 2x + y < 6 d) x + 2y > 7 9. If |x + 1| < 6, then „x‟ belongs to the set a) {x / - 7 < x < 5} b) {x / - 7 < x ≤ 5} c) {x / - 7 ≤ x < 5} d) {x / - 7 ≤ x ≤ 5} 10. x 2 – 4x + 3 < 0, then the value of x lies between: a) 1 and 2 b) 1 and 3 c) 1 and -4 d) 4 and 3 11. If x > 0, y < 0, then (x, y) lies in : From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 172 - a) Q 1 b) Q 2 c) Q 3 d) Q 4 12. The inequation which represents the shaded region is a) 2x + 3y > 6 b) 2x + 3y < 6 c) 3x + 2y < 6 d) 3x + 2y > 6 13. If | -3x | < 6, then a) -3 < x < 3 b) -1 < x < 1 c) -2 < x < 2 d) -2 > x > 2 14. The point which is in the region satisfying the inequation 2x – 3 + 5 < 0 a) (1, 1) b) (-1, 1) c) (2, -1) d) (-1, 2) 15. x = k is the solution of the following inequations : a) x s k ; y s k b) x > k ; y > k c) x s k ; y > k d) x > k ; x s k 16. The graph of the inequality x > 0 is a) + ve Y – axis b) + ve X – axis c) - ve Y – axis d) - ve X – axis 17. If | x | = a (a > 0), then x + a) a b) –a c) a or –a d) a are a 18. If | 5x – 1| > 9, then which of the following belongs to the solution set: a) 1 b) 0 c) 2 d) 3 19. The graph of y = mx 2 is a a) parabola b) hyperbola c) ellipse d) straight line 20. The solution of | x | = -4 a) 4 b) -4 c) 4 or -4 d) no solution 21. If x < 0; y < 0 then (x, y) lies in a) Q 1 b) Q 2 c) Q 3 d) Q 4 22. The figure given below represents a) 3x + 2y > 6 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 173 - b) 3x + 2y s 6 c) 3x + 2y > 6 d) 3x + 2y < 6 23. The value of 2x + 5y should not be less than 75. This is represented by the following inequation a) 2x + 5y < 75 b) 2x + 5y > 75 c) 2x + 5y s 75 d) 2x + 5y > 75 24. Solution of | x – 2 | > 6 a) x < 8 : x > 4 b) x > 8 c) x > 8 : x < -4 d) x > 8 : x > -4 25. If 12 – 3x > 0, then a) x = 4 b) x > 4 c) x < 4 d) x < -4 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 174 - POLYNOM., REMAINDER, & SQUARE ROOTS ____________________________________________________________________________________ 1. a 0 x n + a 1 x n-1 + a 2 x n-2 + …+ a n-1 x + a n is a polynomial in x of the nth degree, if n is a whole number and … = 0. a) a 0 b) a n c) a 1 d) a 2 2. a 0 x n + a 1 x n-1 + a 2 x n-2 + …+ a n is called the zero polynomial if … a) a 0 = a 1 = a 2 = . . . = a n b) a 0 + a 1 + … + a n = 0 c) a 0 = a 1 = a 2 = … = a n = 0 d) a 0 = 0 3. Which of the following is not a polynomial in x? a) x 2 b) 2x 2 + 3x - 5 c) x 2 d) 5x 4. Which of the following is a polynomial in x? a) x -1/2 b) x 1/2 c) 2/x d) x/2 5. f(x) and g(x) are polynomials of the 8 th and the 4 th degrees respectively in x. What is the degree of f(x) / g(x) ? a) 12 b) 4 c) 2 d) any degree less than 8 6. f(x) and g(x) are polynomials of the 10 th and the 2 nd degree respectively. What is the degree of the remainder when f(x) is divided by g(x)? a) 8 b) 2 c) 0 d) 1 or 0 7. What is the remainder when f(x), a rational integral function in x is divided by x – a? a) a b) f(a) c) –f(a) d) f(-a) 8. What is the remainder when f(x) rational integral function in x divided by (ax + b)? a) f(-b) b) f(b/a) c) f(-(b/a)) d) f(-a) 9. If f(x) a rational integral function in x, is divided by (x + a), the remainder is . . . a) f(a) b) f(-a) c) f(1/a) d) f 10. If ax 2 + bx + c is divisible by x-1, then … a) a + b = 0 b) a + c = 0 c) a + b + c = 0 d) a + b = c 11. if f(x) is divided by 4x – 5, the remainder is . . . From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 175 - a) f(4/5) b) f(- 4/5) c) f(5/4) d) f(- 5/4) 12. If f(x) is divided by 5x + 4, the remainder is . . a) f(4/5) b) f(- 4/5) c) f(- 5/4) d) f(5/4) 13. What is the remainder when 3x 2 – 4x + 9 s divided by x + 4? a) 41 b) 73 c) 1 d) 3 14. What is the remainder when x 3 – 3x+ + 3x - 5 is divided by x - 1? a) 4 b) -46 c) -6 15. If the sum of the coefficients of a polynomial in x is zero, then . . . is one of its factors. a) x b) x – 1 c) x + 1 d) x - 2 16. The condition for x + 1 being a factor of ax 3 + bx 2 + cx + d . . . is one of its factors. a) a + b = c + d b) a + b + c + d =0 c) a + b + c = d d) a + c = b + d 17. x 2 – 1 is a factor of ax 4 + bx 3 + cx 2 + dx + e if . . . a) a + b + c + d + e = 0 b) a + c + e = b + d c) a + c + e = b + d = 0 d) a + b = c + d + e 18. If ax 2 -5x+6 is divisible by x – 2, then a = . . . a) 1 b) 2 c) 6 d) 3 19. x – 3 is a factor of 3x 2 – x 2 + px + q if . . . a) p + q = 72 b) b) p + q = 72 c) 3p + q = -72 d) q – 3p = 72 20. Which of the following is a factor of x 3 – 6x 2 + 12x – 8? a) x – 2 b) x + 1 c) x + 2 d) x - 3 21. If f(a/b) = 0, then a factor of f(x) is _____ a) ax – b b) bx – a c) ax + b d) bx + a 22. The sum of the coefficients of the odd power terms of x and the sum of the coefficients of the even power terms of x are equal. Then ____ is a factor of the expression. From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 176 - a) x – 1 b) x c) x + 1 d) x 2 - 1 23. If f(2/3) = 0, then ____ is a factor of f(x). a) 2x – 3 b) 2x + 3 c) 3x – 2 d) 3x + 2 24. One of the factors of x 2 + 19x – 20 is ____ a) x + 1 b) x – 1 c) x – 1 d) x + 2 25. If 2x2 + 9x + k is divisible by x – 3, then k = a) 11 b) 7 c) 9 d) -45 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 177 - QUADRATIC EQUATIONS & EXPRESSIONS ____________________________________________________________________________________ 1. ax 2 + bx + c = 0 is a quadratic polynomial equation in x if a) a = 0 b) b = 0 c) a = b = 0 d) c = 0 2. The number of roots of a quadratic polynomial equation in a single variable is ____ a) 1 b) 3 c) 2 d) infinity 3. The roots of the quadratic equation ax 2 + bx + c = 0 are ____ a) a ac b b a ac b b 2 4 ; 2 4 2 2 ÷ ÷ ÷ ÷ + ÷ b) a ac b b a ac b b 2 4 ; 2 4 2 2 ÷ ÷ ÷ + c) a ac b b a ac b b 2 4 ; 2 4 2 2 ÷ + ÷ ÷ ÷ d) a ac b b a ac b b 2 ; 2 2 2 ÷ ÷ ÷ + + + 4. A root of the equation 13x 2 – 22x – 8 = 0 is a) 3 b) 2 c) 1 d) -2 5. A root of 25x 2 + 57x + 32 = 0 is ____ a) 5 b) 1 c) 2 d) -1 6. One of the roots of the equation x 2 – 2x – 1 = 0 is ____ a) 1 2 + b) 2 3+ c) 1 3 ÷ d) 1 2 ÷ 7. One of the roots of the quadratic equation x 2 + 2x – 1 = 0 is ____ a) 1 2 + b) 1 3 ÷ c) 1 2 ÷ d) 2 3+ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 178 - 8. One of the roots of the equation x 2 – 4x – 1 = 0 is _____ a) 5 2÷ b) 1 2 ÷ c) 2 2÷ d) 2 5 ÷ 9. If o and | are the roots of the equation ax 2 + bx + c = 0, then o + | = _____ a) a b ÷ b) a c c) a b d) a c ÷ 10. If o and | are the roots of the equation px 2 + qx + r = 0, then o | = _____ a) p/r b) –p/r c) r/p d) –r/p 11. If o and | are the roots of the equation ax 2 + bx + c = 0, then o 2 + | 2 = _____ a) b 2 – 4ac b) (b 2 -2ac) / a 2 c) (ab 2 -2c) / a d) (ab 2 +2c) / a 12. If one of the roots of the equation 3x 2 – 2x – k = 0 is 1, then k = ______ a) 1 b) 3 c) -1 d) -2 13. If the roots of the equation ax 2 + bx + c = 0 are a and 1/a, then _____ a) a = c b) b = c c) a = b d) a + c = 0 14. If the roots of the equation x2 + ax + b = 0 are reciprocals of each other, then _____ a) a = 1 b) b = 1 c) +a = -1 d) b = -1 15. The sum of the roots of the equation 3x2 – 5x + 9 = 0 is ______ a) 3 b) -5 c) 5/3 d) -3 16. The sum of the roots of the equation x 2 + ax + b = 0 is 0 if _____ a) a = 0 b) b = 0 c) a = b d) if a or b = 0 17. If one root of 3x 2 – kx + 2 = 0 is 2/3, then k = _____ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 179 - a) 5 b) 1 c) -5 d) -2 18. kx 2 – 5x – 18 = 0 has -2 as one of its roots, then k = _____ a) 1 b) 2 c) 3 d) 4 19. If o and | are the roots of the equation x 2 – px + q = 0 then o 3 | 2 + o 2 | 3 is _____ a) p 2 q b) pq 2 c) pq d) –pq 2 20. If r and s are the roots of the equation ax 2 + bx + c = 0, the value of 1/r 2 + 1/s 2 = _____ a) b 2 – 4ac b) (b 2 – 4ac)/2a c) (b 2 – 4ac)/c 2 d) (b 2 – 2ac)/c 2 21. Sum of the roots of the equation 6x 2 = 1 is _____ a) 1 b) 2 c) 0 d) -4 22. Product of the roots of the equation px 2 + qx – r = 0 is -1, if _____ a) p = r b) p + r = 0 c) q = r d) p = q 23. Sum of all roots of the equation 4x 2 – 8x 2 + 13x – 9 = 0 is ____ a) 8 b) 2 c) -2 d) -8 24. If p and q are the roots of the equation x2 + px + q = 0, then _____ a) p = 1, q = 2 b) p = 2, q = 1 c) p = 1, q = -2 d) p = -2, q = 1 25. x 2 + bx + a = 0 and x2 + ax + b = 0 have a common root; then a) a = b b) a + b = 1 c) a + b + 1 = 0 d) a – b = 1 . From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 180 - RELATIONS AND FUNCTIONS ____________________________________________________________________________________ 1. Which of the following is an ordered pair? a) {5,8} b) 5,8 c) (5,8) d) {(5,8)} 2. A = {p, q, r, s}; B = {c, d}. Which of the following is a relation from A to B? a) {(p, q), (p, c), (q, d), (r, c), (r, s)} b) {p, c), (q, d), (r, c), (s, d)} c) {(r, s), (p, p), (q, d), (s, c)} d) None of the above 3. R is a relation from A to B. Then the domain of the relation is _____ a) A b) B c) a subset A d) a subset of B 4. A = {(-2,8), (3,-4), (5,10), (6,4), (-9,2)}. The domain of a is _____ a) {-2,3,5,6,-9} b) {8,-4,10,4,2} c) {-2,3,4,21} d) {8,3,5,2} 5. P = {(a, q), (I, j), (k, l)}. The range of P is a) {a, i, k} b) {q, j, l} c) {a, j, k} d) (q, j, a) 6. P = {5,6,7}, Q = (6,8). R is a relation from P to Q defined by R = {(x, y)/x < y}. Then R = a) {(5,6), (6,8), (5,8)} b) {(8,7), (6,7), (5,6), (6,8)} c) {(5,6), (5,8), (6,8), (7,8)} d) {(5,5), (6,6), (7,8), (6,8)} 7. A is a subset of B; C is a subset of D. Then a) A x B c C x D b) A x C c B x D c) A x D c B x C d) AC c BD 8. A = {-3,5,8}. Any relation in A cannot contain morethan ____ ordered pairs a) 6 b) 7 c) 8 d) 9 9. B = {11,12,13}. R is a relation in B defined by {(x, y) / (y<x}. Which is it? a) {(11, 12), (11, 13), (12, 13)} b) {(2, 11), (13, 12)} c) {(12, 11), (13, 12), (13, 11)} d) {(12, 11), (12, 13)} 10. A relation R in A is said to be reflexive if _____ a) for every x e A, {(x, x) e R b) for every x, y e A, (x, y) e R ¬ (y, x) e R c) for every x, y e A, (x, y) e R . (y, x) e R ¬ x = y d) for every x, y, z e A, (x, y) e R . (y, z) e R ¬ (x, z) e R From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 181 - 11. If a > b and b>a, then a = b ¬ a, b e R. This relation is _____ a) anti symmetric b) symmetric c) reflexive d) transitive 12. „Greater than‟ in the set of natural numbers is __ relation. a) reflexive b) transitive c) symmetric d) equivalence 13. An example for „transitive‟ relation is ____ a) being subset of b) being ± to in a set of lines in a plane c) being not equal to in a set of numbers d) none of these 14. For a relation to be an equivalence relation it must be a) reflexive b) symmetric c) reflexive, symmetric and transitive d) transitive 15. An example for equivalence relation is ____ a) is a brother of b) is congruent to c) is ± to d) is wife of 16. A relation is a function if no two ordered pairs ____ a) have the same first co-ordinate b) have the some second co-ordinate c) have the some elements d) have a common element 17. Which of the following relations is a function? a) {(a, b), (b, c), (c, d)} b) {(a, b), (a, c), (a, d)} c) {(c, a), (c, b), (c, c)} d) {(c, a), (c, d), (b, d)} 18. Which of the following is a function? a) f = {(1, 2), (1, 3), (1, 4)} b) g = {(1, 2), (2, 3), (3, 4)} c) f = {(x, y), / x = 2, y = (1, 2, 3)} d) g 1 = {a, ) (a, c)} 19. A = {1, 2, 3}; B = {p, q, r, s}, then which of the following is a function from B to A? a) {(q, 1), (p, 2), (r, 3)} b) {(p, 2), (q, 1), (r, 3), (q, 2)} c) {(p, 1), (q, 1), (r, 1) d) {(p, 1), (r, 1), (q, 2), (s, 3)} 20. F: x ÷ 3 Which of the following is correct? a) f(1) = 0 b) f(2) = 0 c) f(3) = 0 d) f(1) = 3 21. f: R ÷ R and g : R ÷ R, f(x) = x + 2, g(x) = 2 – x, then fog and gof a) do not exist b) are equal c) are inverse functions d) none of these 22. R is the set of real numbers. f: R ÷ R, f(x) = x 2 , then f is ______ From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 182 - a) one-one but not onto b) onto but not one-one c) one-one and onto d) Neither one-one nor onto 23. In which of the following functions f(a) = f(-a)? a) x + 2 b) x 2 + 4 c) x 2 + x – 2 d) x 2 + x + 2 24. f is an identity function defined by f(x) = x, then f(12) = _____ a) 0 b) 12 c) 1 d) None of these 25. If a function has its inverse also a function, then it is a) one-one b) one-one and onto c) onto d) one-one and into From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 183 - DERIVATIVES & LIMITS ____________________________________________________________________________________ 1. d/dx (x n ) = a) nx n-1 b) –x.n x-1 c) x/n . n-1 d) xn 2. d/dx(e x ) = a) e x-1 b) e x c) 2e x d) None 3. d/dx (a x ) = a) a x . log a b) x a log x c) x a log a d) a x log x 4. d/dx (log x) = a) 1/x 2 b) x long x c) 1/x d) -1/x 2 5. d/dx (sin x) = a) cos x b) – cos x c) sin 2x d) None 6. d/dx (sin x) = a) –sin x b) sinx c) cos 2 x d) cos 2x 7. d/dx (tan x) = a) 2 sec x b) sec 2 x c) 2 sec 2 x d) secx. tanx 8. d/dx (cot x) = a) –cosec 2 x b) –sec 2 x c) cosec x cos x d) None 9. d/dx (sec x) = a) sec 2 x b) 2secx c) tan 2 x d) sec x tan x 10. d/dx (cosec x) = a) –cosecx cot x b) –cos 2 x c) –cosec 2 x d) 2 sin x 11. d/dx (k) = From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 184 - a) 1 b) k c) 0 d) -1 12. d/dx(1/x 2 ) = a) -1/x 2 b) -2/x 3 c) x d) 2x 2 13. d/dx( x ) = a) x 2 1 b) x 2 1 ÷ c) x 3/2 d) x x 2 14. | | . | \ | x dx d 1 = a) x 2 1 b) x x 2 1 ÷ c) 2 2 1 x d) None 15. d/dx(x 2 – 5x) = a) 2x – 5 b) 2x + 5 c) -2x – 5 d) 5x 2 – 2x 16. d/dx (7x 2 – 5x + 6) = a) 2x – 5 b) 14x – 5 c) 14x 2 – 6 d) 14x 2 + 6 – 5x 17. d/dx (2x 3 – 8x + 4) = a) 6x 2 – 8 b) 6x 2 + 8 c) 8x 2 + 5 d) 3x 2 – 8x + 1 18. d/dx (x 6 – 6x 5 + 5x 2 + 2) = a) 6x 5 + 5x 4 + 2x b) 6x 5 – 30x 4 + 10x c) 6x 5 – 5x 4 – 2x d) 6x 5 + 30x 4 – 10x From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 185 - 19. = | | . | \ | + + 4 5 3 2 2 3 x x dx d a) x x 2 3 15 + b) 3 3 2 2 15 x x + c) x x 3 2 2 15 3 + d) x x 3 2 2 15 2 + 20. = | | . | \ | ÷ x x x dx d 4 3 a) 2 5 6 3 x x + b) 2 5 6 2 1 x x ÷ c) 2 5 5 6 ÷ + x x d) 2 5 6 2 3 ÷ + x x 21. d/dx (x -7 + x 2 – x) = a) 1 2 7 8 ÷ + ÷ x x b) 1 2 7 8 + ÷ x x c) 7x -7 -2x+1 d) None 22. = | | . | \ | x dx d 4 a) -2x -3/2 b) 2/x 3/2 c) x x 2 ÷ d) x x 2 23. = ) 2 ( x dx d From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 186 - a) x 1 ÷ b) x 1 c) x x 1 d) x x 24. = | | . | \ | + + 2 2 5 x x x x dx d a) 3x 2 – 1/x 2 b) 3x 2 + x -2 c) 3x – (1/x 2 ) d) x – (1/3x 2 ) 25. ( ) = x dx d log 2 a) -2/x 2 b) -2/x 3 c) 2/x d) -2/x From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 187 - LOGARITHMS ____________________________________________________________________________________ 1. The logarithmic form of 5 3 = 125 is a) log 5 125 = 3 b) log 3 5 = 125 c) log 125 3 = 5 d) log 3 125 = 5 2. The logarithmic form of 4 3 = 64 is a) log 64 4 = 3 b) log 3 64 = 4 c) log 4 64 = 3 d) log 3 4 = 64 3. The logarithmic form of a b = c is a) log a b = c b) log a c = b c) log b c = a d) log c c = b 4. The Exponential form of log 2 32 = 5 is a) 2 5 = 32 b) 2 32 = 5 c) 5 2 = 32 d) 2 4 = 16 5. The exponential form of log 10 1000 = 3 is a) 10 3 = 1000 b) 10 2 = 0.001 = -3 c) 3 10 = 1000 d) 10 4 = 10000 6. Which of the following is true? a) log 3 81 = 5 b) log 10 0.001 = -3 c) log 12 144 = 3 d) log 2 1/6 = -3 7. log, p + log, q = a) log rq p b) log p qr c) log p pr d) log r pq 8. When (r, s = 0), log p (r/s) = a) log p r-log p s b) log p s-log p r c) log r p-log r s d) log p r-log s p 9. log x x y = a) x log z y b) y log z x c) z log y x d) y log x z 10. log a a= a) 1 b) 0 c) can‟t be determined d) -1 11. log 5 625 = From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 188 - a) 1 b) 2 c) 3 d) 4 12. log 24 = a) log 2 + log 12 b) log 3 + log 8 c) log 4 + log 6 d) All of the above 13. 5 log 2 + 2 log 5 = a) log 800 b) log 54 c) log 49 d) log 60 14. log 1944 = a) 3 log 6 + 2 log 3 b) 6 log 3 + 3 log 2 c) 4 log 3 + 2 log 4 d) 3 log 4 + 5 log 2 15. log 6912 = a) 2 log 4 + 5 log 3 + 2 log 3 b) 4 log 2 + 3 log 3 + 2 log 4 c) 5 log 3 + 3 log 5 + 2 log 6 d) None 16. log 25/81 = a) 2 log 5 – 4 log 3 b) 3 log 4 – 5 log 2 c) 3 log 5 – 2 log 5 d) 3 log 4 + 4 log 5 – log 2 2 17. log 2 + log 6 + log 7 = a) log 84 b) log 72 c) log 96 d) log 108 18. log 3 + log 5 – 3 log 2 = a) log (20/8) b) log (25/9) c) log (15/8) d) log (15/9) 19. log ab/cd = a) log a + log c – log b – log d b) log a + log b – log c – log d c) log c – log a + lob b – log d d) None 20. log 50 1 = a) 1 b) -1 c) 0 d) 2 21. log b y a x = a) x/y log b a b) a/b log y x c) y/x log b a d) a/b log x y 22. = y x x log From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 189 - a) x b) y c) xy d) x/y 23. log a x . log b b. log y c = a) log b a b) log y c c) log y x d) log x y 24. log 5 343 x log 6 5 x log 7 6 = a) 3 b) 4 c) 5 d) 6 25. | . | \ | × 5 3 log b a a) log a 3 + log b 5 b) 3log a + 5 log b c) 3/2 log a + 5/2 log b d) None From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 190 - BINOMIAL THEOREM ____________________________________________________________________________________ 1. n C r = ____ a) n(n – r) b) n(n – 1) _ (n – r) c) n(n – 1)(n – 2) __ (n – r + 1) d) )! ( ! r n r n ÷ 2. 5 C 3 = ____ a) 5 b) 5 × 3 c) 10 d) 5 × 4 × 3 3. The number of terms in the expansion of (x + y) 11 is ____ a) 10 b) 12 c) 11 d) 13 4. If the number of terms in the expansion of n x x | . | \ | ÷ 3 2 3 is 16, the value of n + ____ a) 16 b) 15 c) 17 d) 8 5. The number of terms in the expansion of (3-2x 2 ) 5 is a) 5 b) 6 c) 7 d) 8 6. The (r + 1) th term in the expansion of (a + y) m is ____ a) m C r . a r-m .y r b) m C r . a r y m-r c) m C r a m y r d) m C r a m-r . y r 7. The (r + 1)th term in the expansion of (ax + b) n is ____ a) n C r . x n-r a r b n-r b) n C r . x r a n-r b r c) n C r . a n-r b r x r d) n C r . a n-r b r x n-r 8. The 3 rd term in the expansion of (3x + 5y) 5 is _____ a) 10 x 2 y 2 b) 675 x 3 y 2 c) 6750 x 3 y 2 d) 6750x 3 y 3 9. The 5 th term in the expansion of 6 1 | . | \ | + x x is a) 6 C 4 1/x 2 b) 6 C 5 1/x c) 6 C 2 . x 2 d) 6 C 3 From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 191 - 10. The (r + 1)th term (general term) in the expansion of (x + y) n is ____ a) n C r . x n-r+1 y r+1 b) n C r+1 . x n-r-1 y r c) n C r . xy n-2 b r x r d) n C r . x n-r y r 11. The coefficient of x4 in the expansion of (2x + 3) 7 is _____ a) 7 C 4 . 2 3 .3 4 b) 7 C 4 . 2 4 .3 3 c) 7 C 4 . 2 7 .3 7 d) 7 C 4 . 2 1 .3 7 12. The coefficient of x 2 in the expansion of 6 3 1 2 | . | \ | + x x is _____ a) 6 C 2 . 2 2 . 1/3 4 b) 6 C 2 . 2 4 . 1/3 2 c) 6 C 2 . 2 2 .1/3 2 d) 6 C 2 . 2 4 .1/3 4 13. The (r + 1)th term in the expansion (a + x) m ____ is a) m C r . a r-3 .y r b) m C r a m-r .x r c) m C r a m .x r d) m C r a m-r .x m-r 14. The binomial coefficients successively in the expansion of (x + y) 4 are ____ a) 1, 4, 6 b) 1, 5, 10, 5, 1 c) 1, 3, 3, 1 d) 1, 4, 6, 4, 1 15. The middle term in the expansion of 6 1 | . | \ | + x x a) 6 C 2 b) 6 C 4 c) 6 C 3 d) 6C0× ( ) 5 3 2 3 3 2 b x x x ÷ ÷ | . | \ | ÷ | . | \ | ÷ 16. The middle term in the expansion of 6 2 2 | . | \ | ÷ x x is ____ a) 160x3 b) -160x 3 c) -240x 3 d) 240x 3 17. The middle term in the expansion of (a + b) 4 is _____ a) 6a 2 b 2 b) 4a 3 b 2 c) 6ab 3 d) a 2 b 2 18. The last term in the expansion of 5 2 | . | \ | + x x is _____ a) 2/x 2 b) 10/x 5 c) 32/x 5 d) 32/x From the desk of SOHAIL MERCHANT __________________________________________________________________________________________ - 192 - 19. The middle term in the expansion of 4 2 1 3 | . | \ | + ÷ x x is ______ a) 81x 4 b) 1/16x 4 c) 27/2 d) -54x 2 20. The middle term in the expansion of 8 1 2 | . | \ | ÷ x x is its _____ a) 4 th term b) 5 th term c) 6 th term d) 3 rd term 21. The 4 th term in the expansion of 7 2 2 3 3 4 | | . | \ | ÷ x x is _____ a) 480 x 5 b) -480x 5 c) -325x 5 d) -35.2 5 /3 x 5 22. If the coefficient of x2 in the expansion of (2 + x) n is 240, the value of n = _____ a) 3 b) 4 c) 5 d) 6 23. (1 + x) 4 – (1 – x) 4 = _____ a) 2 + 6x 2 + x 4 b) 2 + 12x 2 + 2x 4 c) 2 + 8x + 12x 2 8x 3 + 2x 4 d) 8x + 8x 3 24. The term independent of x in the expansion of 6 2 | . | \ | + x x is a) 80 b) 6 C 2 c) 160 d) 6 C 3 .2 2